Difference between revisions of "User:Jon Awbrey/SEQUENCES"
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{| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%" | {| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%" | ||
| − | |+ style="height:24px" | <math> | + | |+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math> |
|- style="height:48px; background:#f0f0ff" | |- style="height:48px; background:#f0f0ff" | ||
| <math>\text{Integer}\!</math> | | <math>\text{Integer}\!</math> | ||
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{| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%" | {| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%" | ||
| − | |+ style="height:24px" | <math> | + | |+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math> |
|- style="height:48px; background:#f0f0ff" | |- style="height:48px; background:#f0f0ff" | ||
| <math>\text{Integer}\!</math> | | <math>\text{Integer}\!</math> | ||
| Line 252: | Line 252: | ||
{| align="center" border="1" width="96%" | {| align="center" border="1" width="96%" | ||
| − | |+ style="height: | + | |+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math> |
|- style="height:50px; background:#f0f0ff" | |- style="height:50px; background:#f0f0ff" | ||
| | | | ||
| Line 388: | Line 388: | ||
{| align="center" border="1" width="96%" | {| align="center" border="1" width="96%" | ||
| − | |+ style="height: | + | |+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math> |
|- style="height:50px; background:#f0f0ff" | |- style="height:50px; background:#f0f0ff" | ||
| | | | ||
| Line 1,030: | Line 1,030: | ||
& |P_k| | & |P_k| | ||
\\[10pt] | \\[10pt] | ||
| − | 0 & \{ 1 \} & 1 \\ | + | 0 & \{ 1 \} & 1 |
| − | 1 & \{ 2 \} & 1 \\ | + | \\ |
| − | 2 & \{ 3, 4 \} & 2 \\ | + | 1 & \{ 2 \} & 1 |
| − | 3 & \{ 5, 6, 7, 8, 9, 16 \} & 6 \\ | + | \\ |
| + | 2 & \{ 3, 4 \} & 2 | ||
| + | \\ | ||
| + | 3 & \{ 5, 6, 7, 8, 9, 16 \} & 6 | ||
| + | \\ | ||
4 & \{ 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536 \} & 20 | 4 & \{ 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536 \} & 20 | ||
\end{array}</math> | \end{array}</math> | ||
| Line 1,098: | Line 1,102: | ||
\text{p}_3^1 | \text{p}_3^1 | ||
& = & \text{p}_{\text{p}_2^1}^1 | & = & \text{p}_{\text{p}_2^1}^1 | ||
| − | \\[ | + | \\[12pt] |
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 | & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 | ||
\end{array}</math> | \end{array}</math> | ||
| Line 1,120: | Line 1,124: | ||
\text{p}_4^1 | \text{p}_4^1 | ||
& = & \text{p}_{\text{p}_1^2}^1 | & = & \text{p}_{\text{p}_1^2}^1 | ||
| − | \\[ | + | \\[12pt] |
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 | & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 | ||
\end{array}</math> | \end{array}</math> | ||
| Line 1,132: | Line 1,136: | ||
\text{p}_1^3 | \text{p}_1^3 | ||
& = & \text{p}_1^{\text{p}_2^1} | & = & \text{p}_1^{\text{p}_2^1} | ||
| − | \\[ | + | \\[12pt] |
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1} | & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1} | ||
\end{array}</math> | \end{array}</math> | ||
| Line 1,154: | Line 1,158: | ||
\text{p}_1^4 | \text{p}_1^4 | ||
& = & \text{p}_1^{\text{p}_1^2} | & = & \text{p}_1^{\text{p}_1^2} | ||
| − | \\[ | + | \\[12pt] |
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}} | & = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}} | ||
\end{array}</math> | \end{array}</math> | ||
| Line 1,169: | Line 1,173: | ||
\text{p}_1^1 \text{p}_3^1 | \text{p}_1^1 \text{p}_3^1 | ||
& = & \text{p}_1^1 \text{p}_{\text{p}_2^1}^1 | & = & \text{p}_1^1 \text{p}_{\text{p}_2^1}^1 | ||
| − | \\[ | + | \\[12pt] |
& = & \text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 | & = & \text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 | ||
\end{array}</math> | \end{array}</math> | ||
| Line 1,181: | Line 1,185: | ||
\text{p}_5^1 | \text{p}_5^1 | ||
& = & \text{p}_{\text{p}_3^1}^1 | & = & \text{p}_{\text{p}_3^1}^1 | ||
| − | \\[ | + | \\[12pt] |
& = & \text{p}_{\text{p}_{\text{p}_2^1}^1}^1 | & = & \text{p}_{\text{p}_{\text{p}_2^1}^1}^1 | ||
| − | \\[ | + | \\[12pt] |
& = & \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1 | & = & \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1 | ||
\end{array}</math> | \end{array}</math> | ||
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| [[Image:Riff 12 Big.jpg|65px]] | | [[Image:Riff 12 Big.jpg|65px]] | ||
| [[Image:Rote 12 Big.jpg|105px]] | | [[Image:Rote 12 Big.jpg|105px]] | ||
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|- | |- | ||
| − | | | + | | <math>13\!</math> |
| − | <p>[[Image:Riff | + | | |
| − | < | + | <math>\begin{array}{lll} |
| − | + | \text{p}_6^1 | |
| − | | | + | & = & \text{p}_{\text{p}_1^1 \text{p}_2^1}^1 |
| − | + | \\[12pt] | |
| − | < | + | & = & \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}^1 |
| − | + | \end{array}</math> | |
| − | | | + | | <math>\text{p}_{\text{p} \text{p}_{\text{p}}}\!</math> |
| − | + | | [[Image:Riff 13 Big.jpg|65px]] | |
| − | < | + | | [[Image:Rote 13 Big.jpg|80px]] |
| − | + | |- | |
| − | | | + | | <math>14\!</math> |
| − | + | | | |
| − | < | + | <math>\begin{array}{lll} |
| − | + | \text{p}_1^1 \text{p}_4^1 | |
| − | | | + | & = & \text{p}_1^1 \text{p}_{\text{p}_1^2}^1 |
| − | + | \\[12pt] | |
| − | + | & = & \text{p}_1^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 | |
| − | < | + | \end{array}</math> |
| + | | <math>\text{p} \text{p}_{\text{p}^{\text{p}}}\!</math> | ||
| + | | [[Image:Riff 14 Big.jpg|90px]] | ||
| + | | [[Image:Rote 14 Big.jpg|105px]] | ||
| + | |- | ||
| + | | <math>17\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_7^1 | ||
| + | & = & \text{p}_{\text{p}_4^1}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_{\text{p}_1^2}^1}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1 | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!</math> | ||
| + | | [[Image:Riff 17 Big.jpg|90px]] | ||
| + | | [[Image:Rote 17 Big.jpg|65px]] | ||
| + | |- | ||
| + | | <math>18\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_1^1 \text{p}_2^2 | ||
| + | & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p} \text{p}_{\text{p}}^{\text{p}}\!</math> | ||
| + | | [[Image:Riff 18 Big.jpg|65px]] | ||
| + | | [[Image:Rote 18 Big.jpg|120px]] | ||
| + | |- | ||
| + | | <math>19\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_8^1 | ||
| + | & = & \text{p}_{\text{p}_1^3}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_1^{\text{p}_2^1}}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}^1 | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!</math> | ||
| + | | [[Image:Riff 19 Big.jpg|90px]] | ||
| + | | [[Image:Rote 19 Big.jpg|65px]] | ||
| + | |- | ||
| + | | <math>23\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_9^1 | ||
| + | & = & \text{p}_{\text{p}_2^2}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^1 | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}_{\text{p}_{\text{p}}^{\text{p}}}\!</math> | ||
| + | | [[Image:Riff 23 Big.jpg|65px]] | ||
| + | | [[Image:Rote 23 Big.jpg|80px]] | ||
| + | |- | ||
| + | | <math>25\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_3^2 | ||
| + | & = & \text{p}_{\text{p}_2^1}^{\text{p}_1^1} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^{\text{p}_1^1} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}_{\text{p}_{\text{p}}}^{\text{p}}\!</math> | ||
| + | | [[Image:Riff 25 Big.jpg|65px]] | ||
| + | | [[Image:Rote 25 Big.jpg|80px]] | ||
| + | |- | ||
| + | | <math>27\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_2^3 | ||
| + | & = & \text{p}_{\text{p}_1^1}^{\text{p}_2^1} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_1^1}^{\text{p}_{\text{p}_1^1}^1} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}_{\text{p}}^{\text{p}_{\text{p}}}\!</math> | ||
| + | | [[Image:Riff 27 Big.jpg|65px]] | ||
| + | | [[Image:Rote 27 Big.jpg|80px]] | ||
| + | |- | ||
| + | | <math>32\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_1^5 | ||
| + | & = & \text{p}_1^{\text{p}_3^1} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^{\text{p}_{\text{p}_2^1}^1} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}^{\text{p}_{\text{p}_{\text{p}}}}\!</math> | ||
| + | | [[Image:Riff 32 Big.jpg|90px]] | ||
| + | | [[Image:Rote 32 Big.jpg|65px]] | ||
| + | |- | ||
| + | | <math>49\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_4^2 | ||
| + | & = & \text{p}_{\text{p}_1^2}^{\text{p}_1^1} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^{\text{p}_1^1} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}_{\text{p}^{\text{p}}}^{\text{p}}\!</math> | ||
| + | | [[Image:Riff 49 Big.jpg|65px]] | ||
| + | | [[Image:Rote 49 Big.jpg|80px]] | ||
| + | |- | ||
| + | | <math>53\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_{16}^1 | ||
| + | & = & \text{p}_{\text{p}_1^4}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_1^{\text{p}_1^2}}^1 | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1 | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}_{\text{p}^{\text{p}^{\text{p}}}}\!</math> | ||
| + | | [[Image:Riff 53 Big.jpg|90px]] | ||
| + | | [[Image:Rote 53 Big.jpg|90px]] | ||
|- | |- | ||
| − | | | + | | <math>64\!</math> |
| − | < | + | | |
| − | + | <math>\begin{array}{lll} | |
| − | + | \text{p}_1^6 | |
| − | | | + | & = & \text{p}_1^{\text{p}_1^1 \text{p}_2^1} |
| − | + | \\[12pt] | |
| − | + | & = & \text{p}_1^{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1} | |
| − | + | \end{array}</math> | |
| − | | | + | | <math>\text{p}^{\text{p} \text{p}_{\text{p}}}\!</math> |
| − | + | | [[Image:Riff 64 Big.jpg|65px]] | |
| − | + | | [[Image:Rote 64 Big.jpg|105px]] | |
| − | + | |- | |
| − | | | + | | <math>81\!</math> |
| − | + | | | |
| − | + | <math>\begin{array}{lll} | |
| − | + | \text{p}_2^4 | |
| − | | | + | & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^2} |
| − | + | \\[12pt] | |
| − | + | & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^{\text{p}_1^1}} | |
| − | + | \end{array}</math> | |
| + | | <math>\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\!</math> | ||
| + | | [[Image:Riff 81 Big.jpg|65px]] | ||
| + | | [[Image:Rote 81 Big.jpg|105px]] | ||
| + | |- | ||
| + | | <math>128\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_1^7 | ||
| + | & = & \text{p}_1^{\text{p}_4^1} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^{\text{p}_{\text{p}_1^2}^1} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\!</math> | ||
| + | | [[Image:Riff 128 Big.jpg|90px]] | ||
| + | | [[Image:Rote 128 Big.jpg|90px]] | ||
|- | |- | ||
| − | | | + | | <math>256\!</math> |
| − | < | + | | |
| − | + | <math>\begin{array}{lll} | |
| − | + | \text{p}_1^8 | |
| − | + | & = & \text{p}_1^{\text{p}_1^3} | |
| − | + | \\[12pt] | |
| − | + | & = & \text{p}_1^{\text{p}_1^{\text{p}_2^1}} | |
| − | + | \\[12pt] | |
| − | + | & = & \text{p}_1^{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}} | |
| − | + | \end{array}</math> | |
| − | + | | <math>\text{p}^{\text{p}^{\text{p}_{\text{p}}}}\!</math> | |
| − | + | | [[Image:Riff 256 Big.jpg|90px]] | |
| − | + | | [[Image:Rote 256 Big.jpg|90px]] | |
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|- | |- | ||
| − | | | + | | <math>512\!</math> |
| − | < | + | | |
| − | + | <math>\begin{array}{lll} | |
| − | + | \text{p}_1^9 | |
| − | + | & = & \text{p}_1^{\text{p}_2^2} | |
| − | + | \\[12pt] | |
| − | + | & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}} | |
| − | + | \end{array}</math> | |
| − | | | + | | <math>\text{p}^{\text{p}_{\text{p}}^{\text{p}}}\!</math> |
| − | + | | [[Image:Riff 512 Big.jpg|65px]] | |
| − | + | | [[Image:Rote 512 Big.jpg|105px]] | |
| − | |||
| − | | | ||
| − | |||
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|- | |- | ||
| + | | <math>65536\!</math> | ||
| + | | | ||
| + | <math>\begin{array}{lll} | ||
| + | \text{p}_1^{16} | ||
| + | & = & \text{p}_1^{\text{p}_1^4} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^{\text{p}_1^{\text{p}_1^2}} | ||
| + | \\[12pt] | ||
| + | & = & \text{p}_1^{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}} | ||
| + | \end{array}</math> | ||
| + | | <math>\text{p}^{\text{p}^{\text{p}^{\text{p}}}}\!</math> | ||
| + | | [[Image:Riff 65536 Big.jpg|90px]] | ||
| + | | [[Image:Rote 65536 Big.jpg|115px]] | ||
| + | |} | ||
| + | |} | ||
| + | |||
| + | ===ASCII=== | ||
| + | |||
| + | <pre> | ||
| + | Example | ||
| + | |||
| + | * k | natural numbers n such that |riff(n)| = k | ||
| + | * 0 | 1; | ||
| + | * 1 | 2; | ||
| + | * 2 | 3, 4; | ||
| + | * 3 | 5, 6, 7, 8, 9, 16; | ||
| + | * 4 | 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536; | ||
| + | * The natural number values for the riffs with at most 3 pts are as follows (x = root): | ||
| + | * .................o.......o..o.......o | ||
| + | * .................|.......^..|.......^ | ||
| + | * .................v.......|..v.......| | ||
| + | * ...........o..o..o....o..o..o..o.o..o | ||
| + | * ...........|..^..|....|..|..^..|.^..^ | ||
| + | * ...........v..|..v....v..v..|..v/...| | ||
| + | * Riff:...x;.x,.x;.x,.x.x,.x,.x,.x,...x; | ||
| + | * Value:..2;.3,.4;.5,..6.,.7,.8,.9,..16; | ||
| + | </pre> | ||
| + | |||
| + | ==A062537== | ||
| + | |||
| + | * [http://oeis.org/wiki/A062537 A062537] | ||
| + | |||
| + | ===Wiki + TeX + JPEG=== | ||
| + | |||
| + | {| align="center" border="1" cellpadding="10" | ||
| + | |+ style="height:25px" | <math>a(n) = \text{Number of Nodes in the Riff of}~ n</math> | ||
| valign="bottom" | | | valign="bottom" | | ||
| − | <p> | + | <p> </p><br> |
| − | <p><math> | + | <p><math>1\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(1) ~=~ 0</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 2 Big.jpg|20px]]</p><br> |
| − | <p><math>\text{p | + | <p><math>\text{p}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(2) ~=~ 1</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 3 Big.jpg|40px]]</p><br> |
| − | <p><math> | + | <p><math>\text{p}_\text{p}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(3) ~=~ 2</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 4 Big.jpg|40px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p}^\text{p}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(4) ~=~ 2</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 5 Big.jpg|65px]]</p><br> |
| − | <p><math> | + | <p><math>\text{p}_{\text{p}_{\text{p}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(5) ~=~ 3</math></p> |
|- | |- | ||
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 6 Big.jpg|65px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p} \text{p}_{\text{p}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(6) ~=~ 3</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 7 Big.jpg|65px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p}_{\text{p}^{\text{p}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(7) ~=~ 3</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 8 Big.jpg|65px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p}^{\text{p}_{\text{p}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(8) ~=~ 3</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 9 Big.jpg|40px]]</p><br> |
| − | <p><math> | + | <p><math>\text{p}_\text{p}^\text{p}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(9) ~=~ 3</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 10 Big.jpg|90px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(10) ~=~ 4</math></p> |
|- | |- | ||
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 11 Big.jpg|90px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(11) ~=~ 4</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 12 Big.jpg|65px]]</p><br> |
| − | <p><math> | + | <p><math>\text{p}^\text{p} \text{p}_\text{p}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(12) ~=~ 4</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 13 Big.jpg|65px]]</p><br> |
| − | <p><math> | + | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(13) ~=~ 4</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 14 Big.jpg|90px]]</p><br> |
| − | <p><math> | + | <p><math>\text{p} \text{p}_{\text{p}^{\text{p}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(14) ~=~ 4</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 15 Big.jpg|90px]]</p><br> |
| − | <p><math> | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(15) ~=~ 5</math></p> |
|- | |- | ||
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 16 Big.jpg|65px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p}^{\text{p}^{\text{p}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(16) ~=~ 3</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 17 Big.jpg|90px]]</p><br> |
| − | <p><math> | + | <p><math>\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(17) ~=~ 4</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 18 Big.jpg|65px]]</p><br> |
| − | <p><math> | + | <p><math>\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(18) ~=~ 4</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 19 Big.jpg|90px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(19) ~=~ 4</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 20 Big.jpg|90px]]</p><br> |
| − | <p><math> | + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(20) ~=~ 5</math></p> |
|- | |- | ||
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 21 Big.jpg|90px]]</p><br> |
| − | <p><math>\text{p} \text{p} | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(21) ~=~ 5</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 22 Big.jpg|115px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(22) ~=~ 5</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 23 Big.jpg|65px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(23) ~=~ 4</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 24 Big.jpg|115px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(24) ~=~ 5</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 25 Big.jpg|65px]]</p><br> |
| − | <p><math> | + | <p><math>\text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(25) ~=~ 4</math></p> |
|- | |- | ||
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 26 Big.jpg|90px]]</p><br> |
| − | <p><math> | + | <p><math>\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(26) ~=~ 5</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 27 Big.jpg|65px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(27) ~=~ 4</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 28 Big.jpg|90px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(28) ~=~ 5</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 29 Big.jpg|90px]]</p><br> |
| − | <p><math>\text{p} \text{p} | + | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(29) ~=~ 5</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 30 Big.jpg|115px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(30) ~=~ 6</math></p> |
|- | |- | ||
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 31 Big.jpg|115px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(31) ~=~ 5</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 32 Big.jpg|90px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p}^{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(32) ~=~ 4</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 33 Big.jpg|115px]]</p><br> |
| − | <p><math>\text{p} \text{p} | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(33) ~=~ 6</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 34 Big.jpg|115px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(34) ~=~ 5</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 35 Big.jpg|90px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(35) ~=~ 6</math></p> |
| − | | | + | |- |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| valign="bottom" | | | valign="bottom" | | ||
| − | <p> | + | <p>[[Image:Riff 36 Big.jpg|65px]]</p><br> |
| − | <p><math> | + | <p><math>\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(36) ~=~ 5</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 37 Big.jpg|65px]]</p><br> |
| − | <p><math>\text{p}\!</math></p><br> | + | <p><math>\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(37) ~=~ 5</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 38 Big.jpg|115px]]</p><br> |
| − | <p><math>\text{p}_\text{p}\!</math></p><br> | + | <p><math>\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(38) ~=~ 5</math></p> |
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 39 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(39) ~=~ 6</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 40 Big.jpg|135px]]</p><br> |
| − | <p><math>\text{p}_{\text{p}_{\text{p}}}\!</math></p><br> | + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(40) ~=~ 6</math></p> |
| + | |- | ||
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 41 Big.jpg|90px]]</p><br> |
| − | <p><math>\text{p} \text{p}_{\text{p}_ | + | <p><math>\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(41) ~=~ 5</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 42 Big.jpg|115px]]</p><br> |
| − | <p><math>\text{p}_\text{p} \text{p}_{\text{p} | + | <p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(42) ~=~ 6</math></p> |
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image:Riff | + | <p>[[Image:Riff 43 Big.jpg|90px]]</p><br> |
| − | <p><math>\text{p} | + | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!</math></p><br> |
| − | <p><math>a( | + | <p><math>a(43) ~=~ 5</math></p> |
| − | | | + | | valign="bottom" | |
| − | + | <p>[[Image:Riff 44 Big.jpg|115px]]</p><br> | |
| − | + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | |
| − | + | <p><math>a(44) ~=~ 6</math></p> | |
| − | + | | valign="bottom" | | |
| − | + | <p>[[Image:Riff 45 Big.jpg|90px]]</p><br> | |
| − | + | <p><math>\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> | |
| − | + | <p><math>a(45) ~=~ 6</math></p> | |
| − | + | |- | |
| − | + | | valign="bottom" | | |
| − | <math>\ | + | <p>[[Image:Riff 46 Big.jpg|90px]]</p><br> |
| − | + | <p><math>\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br> | |
| − | + | <p><math>a(46) ~=~ 5</math></p> | |
| − | + | | valign="bottom" | | |
| − | + | <p>[[Image:Riff 47 Big.jpg|90px]]</p><br> | |
| − | + | <p><math>\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> | |
| − | + | <p><math>a(47) ~=~ 6</math></p> | |
| − | + | | valign="bottom" | | |
| − | + | <p>[[Image:Riff 48 Big.jpg|65px]]</p><br> | |
| − | + | <p><math>\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!</math></p><br> | |
| − | + | <p><math>a(48) ~=~ 5</math></p> | |
| − | + | | valign="bottom" | | |
| − | + | <p>[[Image:Riff 49 Big.jpg|65px]]</p><br> | |
| − | + | <p><math>\text{p}_{\text{p}^\text{p}}^\text{p}\!</math></p><br> | |
| − | + | <p><math>a(49) ~=~ 4</math></p> | |
| − | + | | valign="bottom" | | |
| − | + | <p>[[Image:Riff 50 Big.jpg|90px]]</p><br> | |
| − | + | <p><math>\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br> | |
| − | + | <p><math>a(50) ~=~ 5</math></p> | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | = | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
|- | |- | ||
| − | | || || || || || | | + | | valign="bottom" | |
| − | + | <p>[[Image:Riff 51 Big.jpg|115px]]</p><br> | |
| − | | || | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br> |
| − | + | <p><math>a(51) ~=~ 6</math></p> | |
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 52 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(52) ~=~ 6</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 53 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}^{\text{p}^\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(53) ~=~ 4</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 54 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(54) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 55 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(55) ~=~ 7</math></p> | ||
|- | |- | ||
| − | | || | + | | valign="bottom" | |
| − | + | <p>[[Image:Riff 56 Big.jpg|135px]]</p><br> | |
| − | | || | + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| − | + | <p><math>a(56) ~=~ 6</math></p> | |
| − | | | + | | valign="bottom" | |
| − | | | + | <p>[[Image:Riff 57 Big.jpg|115px]]</p><br> |
| − | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br> | |
| − | | || | + | <p><math>a(57) ~=~ 6</math></p> |
| − | + | | valign="bottom" | | |
| − | | | + | <p>[[Image:Riff 58 Big.jpg|115px]]</p><br> |
| − | | | + | <p><math>\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| − | + | <p><math>a(58) ~=~ 6</math></p> | |
| − | | | | + | | valign="bottom" | |
| − | | | + | <p>[[Image:Riff 59 Big.jpg|115px]]</p><br> |
| − | | | + | <p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!</math></p><br> |
| − | + | <p><math>a(59) ~=~ 5</math></p> | |
| − | + | | valign="bottom" | | |
| − | + | <p>[[Image:Riff 60 Big.jpg|115px]]</p><br> | |
| − | + | <p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> | |
| − | | | + | <p><math>a(60) ~=~ 7</math></p> |
| − | | | + | |} |
| − | + | ||
| − | + | ==A062860== | |
| − | + | ||
| − | | | + | * [http://oeis.org/wiki/A062860 A062860] |
| − | | | + | |
| − | + | ===Wiki + TeX + JPEG=== | |
| − | | || | + | |
| − | + | {| align="center" border="1" cellpadding="10" | |
| + | |+ style="height:25px" | <math>a(n) = \text{Least Integer}~ j ~\text{with}~ n ~\text{Nodes in Its Riff}</math> | ||
| + | | valign="bottom" | | ||
| + | <p> </p><br> | ||
| + | <p><math>1\!</math></p><br> | ||
| + | <p><math>a(0) ~=~ 1</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 2 Big.jpg|20px]]</p><br> | ||
| + | <p><math>\text{p}\!</math></p><br> | ||
| + | <p><math>a(1) ~=~ 2</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 3 Big.jpg|40px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p}\!</math></p><br> | ||
| + | <p><math>a(2) ~=~ 3</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 5 Big.jpg|65px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_{\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(3) ~=~ 5</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 10 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(4) ~=~ 10</math></p> | ||
|- | |- | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 15 Big.jpg|90px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(5) ~=~ 15</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 30 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> | ||
| + | <p><math>a(6) ~=~ 30</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 55 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(7) ~=~ 55</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 105 Big.jpg|115px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> | ||
| + | <p><math>a(8) ~=~ 105</math></p> | ||
| + | | valign="bottom" | | ||
| + | <p>[[Image:Riff 165 Big.jpg|135px]]</p><br> | ||
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | ||
| + | <p><math>a(9) ~=~ 165</math></p> | ||
| + | |} | ||
| + | |||
| + | ==A106177== | ||
| + | |||
| + | * [http://oeis.org/wiki/A106177 A106177] | ||
| + | |||
| + | ===Primal Codes of Finite Partial Functions on Positive Integers=== | ||
| + | |||
| + | {| align="center" | ||
| | | | ||
| − | + | <math>\begin{array}{rcl} | |
| − | + | 1 & = & \varnothing \\ | |
| − | + | 2 & = & 1\!:\!1 \\ | |
| − | + | 3 & = & 2\!:\!1 \\ | |
| − | + | 4 & = & 1\!:\!2 \\ | |
| − | + | 5 & = & 3\!:\!1 \\ | |
| − | + | 6 & = & 1\!:\!1 ~~ 2\!:\!1 \\ | |
| − | + | 7 & = & 4\!:\!1 \\ | |
| − | + | 8 & = & 1\!:\!3 \\ | |
| − | + | 9 & = & 2\!:\!2 \\ | |
| − | + | 10 & = & 1\!:\!1 ~~ 3\!:\!1 \\ | |
| − | + | 11 & = & 5\!:\!1 \\ | |
| − | + | 12 & = & 1\!:\!2 ~~ 2\!:\!1 \\ | |
| − | + | 13 & = & 6\!:\!1 \\ | |
| − | + | 14 & = & 1\!:\!1 ~~ 4\!:\!1 \\ | |
| − | + | 15 & = & 2\!:\!1 ~~ 3\!:\!1 \\ | |
| − | + | 16 & = & 1\!:\!4 \\ | |
| − | + | 17 & = & 7\!:\!1 \\ | |
| − | + | 18 & = & 1\!:\!1 ~~ 2\!:\!2 \\ | |
| − | + | 19 & = & 8\!:\!1 \\ | |
| − | + | 20 & = & 1\!:\!2 ~~ 3\!:\!1 | |
| − | + | \end{array}</math> | |
| − | |||
| − | |||
| − | |||
|} | |} | ||
| − | ===Wiki | + | ===Wiki Table=== |
| − | == | + | {| align="center" style="font-weight:bold; text-align:center" |
| − | + | | || || || || || || || || | |
| − | + | | <font color="red">1</font> | |
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| − | < | + | | <font color="red">1</font> |
| − | + | |- | |
| − | + | | || || || || || || || | |
| − | + | | <font color="red">2</font> | |
| − | + | | || 1 || | |
| − | + | | <font color="red">2</font> | |
| − | + | |- | |
| − | + | | || || || || || || | |
| − | + | | <font color="red">3</font> | |
| − | + | | || 1 || || 1 || | |
| − | + | | <font color="red">3</font> | |
| − | + | |- | |
| − | + | | || || || || || | |
| − | + | | <font color="red">4</font> | |
| − | + | | || 1 || || 2 || || 1 || | |
| − | + | | <font color="red">4</font> | |
| − | + | |- | |
| − | + | | || || || || | |
| − | + | | <font color="red">5</font> | |
| − | + | | || 1 || || 3 || || 1 || || 1 || | |
| − | + | | <font color="red">5</font> | |
| − | | | + | |- |
| − | + | | || || || | |
| − | + | | <font color="red">6</font> | |
| − | + | | || 1 || || 1 || || 1 || || 4 || || 1 || | |
| − | + | | <font color="red">6</font> | |
| − | + | |- | |
| − | < | + | | || || |
| − | + | | <font color="red">7</font> | |
| − | + | | || 1 || || 5 || || 2 || || 9 || || 1 || || 1 || | |
| − | + | | <font color="red">7</font> | |
| − | + | |- | |
| − | + | | || | |
| − | + | | <font color="red">8</font> | |
| − | + | | || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || | |
| − | + | | <font color="red">8</font> | |
| − | + | |- | |
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| − | |||
| − | |||
| | | | ||
| − | <math>\begin{ | + | | <font color="red">9</font> |
| − | + | | || 1 || || 7 || || 1 || || 25|| || 1 || || 3 || || 1 || || 1 || | |
| + | | <font color="red">9</font> | ||
| + | |- | ||
| + | | width="12pt" | <font color="red">10</font> | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | 1 | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | 1 | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | 1 | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | 36 | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | 1 | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | 2 | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | 1 | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | 8 | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | 1 | ||
| + | | width="12pt" | | ||
| + | | width="12pt" | <font color="red">10</font> | ||
| + | |} | ||
| + | |||
| + | ===Wiki + TeX=== | ||
| + | |||
| + | ====Smallmatrix==== | ||
| + | |||
| + | {| align="center" | ||
| + | | | ||
| + | <math>\begin{smallmatrix} | ||
| + | & & & & & & & & & {\color{red}1} & & {\color{red}1} | ||
\\ | \\ | ||
| − | 1 | + | & & & & & & & & {\color{red}2} & & 1 & & {\color{red}2} |
\\ | \\ | ||
| − | + | & & & & & & & {\color{red}3} & & 1 & & 1 & & {\color{red}3} | |
\\ | \\ | ||
| − | + | & & & & & & {\color{red}4} & & 1 & & 2 & & 1 & & {\color{red}4} | |
\\ | \\ | ||
| − | + | & & & & & {\color{red}5} & & 1 & & 3 & & 1 & & 1 & & {\color{red}5} | |
\\ | \\ | ||
| − | + | & & & & {\color{red}6} & & 1 & & 1 & & 1 & & 4 & & 1 & & {\color{red}6} | |
\\ | \\ | ||
| − | + | & & & {\color{red}7} & & 1 & & 5 & & 2 & & 9 & & 1 & & 1 & & {\color{red}7} | |
\\ | \\ | ||
| − | + | & & {\color{red}8} & & 1 & & 6 & & 1 & & 1 & & 1 & & 2 & & 1 & & {\color{red}8} | |
\\ | \\ | ||
| − | + | & {\color{red}9} & & 1 & & 7 & & 1 & & 25 & & 1 & & 3 & & 1 & & 1 & & {\color{red}9} | |
\\ | \\ | ||
| − | + | {\color{red}10} & & 1 & & 1 & & 1 & & 36 & & 1 & & 2 & & 1 & & 8 & & 1 & & {\color{red}10} | |
| − | + | \end{smallmatrix}</math> | |
| − | 10 | ||
| − | \end{ | ||
|} | |} | ||
| − | === | + | ====Array==== |
| − | < | + | {| align="center" |
| − | + | | | |
| − | + | <math>\begin{array}{*{21}{c}} | |
| − | + | & & & & & & & & & {\color{red}1} & & {\color{red}1} | |
| − | + | \\ | |
| − | + | & & & & & & & & {\color{red}2} & & 1 & & {\color{red}2} | |
| − | + | \\ | |
| − | + | & & & & & & & {\color{red}3} & & 1 & & 1 & & {\color{red}3} | |
| − | + | \\ | |
| − | + | & & & & & & {\color{red}4} & & 1 & & 2 & & 1 & & {\color{red}4} | |
| − | + | \\ | |
| − | + | & & & & & {\color{red}5} & & 1 & & 3 & & 1 & & 1 & & {\color{red}5} | |
| − | + | \\ | |
| − | + | & & & & {\color{red}6} & & 1 & & 1 & & 1 & & 4 & & 1 & & {\color{red}6} | |
| − | + | \\ | |
| − | + | & & & {\color{red}7} & & 1 & & 5 & & 2 & & 9 & & 1 & & 1 & & {\color{red}7} | |
| − | + | \\ | |
| − | + | & & {\color{red}8} & & 1 & & 6 & & 1 & & 1 & & 1 & & 2 & & 1 & & {\color{red}8} | |
| − | + | \\ | |
| − | + | & {\color{red}9} & & 1 & & 7 & & 1 & & 25 & & 1 & & 3 & & 1 & & 1 & & {\color{red}9} | |
| − | + | \\ | |
| − | + | {\color{red}10} & & 1 & & 1 & & 1 & & 36 & & 1 & & 2 & & 1 & & 8 & & 1 & & {\color{red}10} | |
| − | + | \end{array}</math> | |
| − | + | |} | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
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| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | </ | ||
| − | == | + | ====Matrix==== |
| − | + | {| align="center" | |
| − | + | | | |
| − | === | + | <math>\begin{matrix} |
| + | n \circ m | ||
| + | \\ | ||
| + | 1 ~/~\backslash~ 1 | ||
| + | \\ | ||
| + | 2 ~/~ 1 ~\backslash~ 2 | ||
| + | \\ | ||
| + | 3 ~/~ 1 \cdot 1 ~\backslash~ 3 | ||
| + | \\ | ||
| + | 4 ~/~ 1 \cdot 2 \cdot 1 ~\backslash~ 4 | ||
| + | \\ | ||
| + | 5 ~/~ 1 \cdot 3 \cdot 1 \cdot 1 ~\backslash~ 5 | ||
| + | \\ | ||
| + | 6 ~/~ 1 \cdot 1 \cdot 1 \cdot 4 \cdot 1 ~\backslash~ 6 | ||
| + | \\ | ||
| + | 7 ~/~ 1 \cdot 5 \cdot 2 \cdot 9 \cdot 1 \cdot 1 ~\backslash~ 7 | ||
| + | \\ | ||
| + | 8 ~/~ 1 \cdot 6 \cdot 1 \cdot 1 \cdot 1 \cdot 2 \cdot 1 ~\backslash~ 8 | ||
| + | \\ | ||
| + | 9 ~/~ 1 \cdot 7 \cdot 1 \cdot 25\cdot 1 \cdot 3 \cdot 1 \cdot 1 ~\backslash~ 9 | ||
| + | \\ | ||
| + | 10 ~/~ 1 \cdot 1 \cdot 1 \cdot 36\cdot 1 \cdot 2 \cdot 1 \cdot 8 \cdot 1 ~\backslash~ 10 | ||
| + | \end{matrix}</math> | ||
| + | |} | ||
| + | |||
| + | ===ASCII=== | ||
| − | + | <pre> | |
| − | + | Example | |
| − | + | ||
| − | + | * n o m | |
| − | + | * \ / | |
| − | + | * 1 . 1 | |
| − | + | * \ / \ / | |
| − | + | * 2 . 1 . 2 | |
| − | + | * \ / \ / \ / | |
| − | + | * 3 . 1 . 1 . 3 | |
| − | + | * \ / \ / \ / \ / | |
| − | + | * 4 . 1 . 2 . 1 . 4 | |
| − | + | * \ / \ / \ / \ / \ / | |
| − | + | * 5 . 1 . 3 . 1 . 1 . 5 | |
| − | + | * \ / \ / \ / \ / \ / \ / | |
| − | + | * 6 . 1 . 1 . 1 . 4 . 1 . 6 | |
| − | + | * \ / \ / \ / \ / \ / \ / \ / | |
| − | + | * 7 . 1 . 5 . 2 . 9 . 1 . 1 . 7 | |
| − | + | * \ / \ / \ / \ / \ / \ / \ / \ / | |
| − | + | * 8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 8 | |
| − | + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / | |
| − | + | * 9 . 1 . 7 . 1 . 25. 1 . 3 . 1 . 1 . 9 | |
| − | + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / | |
| − | + | * 10 . 1 . 1 . 1 . 36. 1 . 2 . 1 . 8 . 1 . 10 | |
| − | + | * | |
| − | + | * Primal codes of finite partial functions on positive integers: | |
| − | + | * 1 = { } | |
| − | + | * 2 = 1:1 | |
| − | + | * 3 = 2:1 | |
| − | + | * 4 = 1:2 | |
| − | + | * 5 = 3:1 | |
| − | + | * 6 = 1:1 2:1 | |
| − | | | + | * 7 = 4:1 |
| − | | | | + | * 8 = 1:3 |
| − | | | + | * 9 = 2:2 |
| − | | | + | * 10 = 1:1 3:1 |
| − | | || || || || || || || | + | * 11 = 5:1 |
| − | | <font color="red"> | + | * 12 = 1:2 2:1 |
| − | + | * 13 = 6:1 | |
| − | | <font color="red"> | + | * 14 = 1:1 4:1 |
| + | * 15 = 2:1 3:1 | ||
| + | * 16 = 1:4 | ||
| + | * 17 = 7:1 | ||
| + | * 18 = 1:1 2:2 | ||
| + | * 19 = 8:1 | ||
| + | * 20 = 1:2 3:1 | ||
| + | </pre> | ||
| + | |||
| + | ==A106178== | ||
| + | |||
| + | * [http://oeis.org/wiki/A106178 A106178] | ||
| + | |||
| + | ===Wiki Table=== | ||
| + | |||
| + | {| align="center" style="font-weight:bold; text-align:center; width:90%" | ||
| + | | || || || || || || || || || || || || || || | ||
| + | | <font color="red">1</font> | ||
| + | | | ||
| + | | <font color="red">1</font> | ||
|- | |- | ||
| − | | || || || || || || | + | | || || || || || || || || || || || || || |
| − | | <font color="red"> | + | | <font color="red">2</font> |
| − | + | | || · || | |
| − | | <font color="red"> | + | | <font color="red">2</font> |
|- | |- | ||
| − | | || || || || || | + | | || || || || || || || || || || || || |
| − | | <font color="red"> | + | | <font color="red">3</font> |
| − | | || · | + | | || · || || · || |
| − | | <font color="red"> | + | | <font color="red">3</font> |
|- | |- | ||
| − | | || || || || | + | | || || || || || || || || || || || |
| − | | <font color="red"> | + | | <font color="red">4</font> |
| − | | || · || || | + | | || · || || 2 || || · || |
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|- | |- | ||
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| − | | <font color="red"> | + | | <font color="red">6</font> |
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| − | | <font color="red"> | + | | <font color="red">7</font> |
| − | | || · || || | + | | || · || || 5 || || 2 || || 9 || || 1 || || · || |
| − | | <font color="red"> | + | | <font color="red">7</font> |
| + | |- | ||
| + | | || || || || || || || | ||
| + | | <font color="red">8</font> | ||
| + | | || · || || 6 || || 1 || || 1 || || 1 || || 2 || || · || | ||
| + | | <font color="red">8</font> | ||
| + | |- | ||
| + | | || || || || || || | ||
| + | | <font color="red">9</font> | ||
| + | | || · || || 7 || || 1 || || 25|| || 1 || || 3 || || 1 || || · || | ||
| + | | <font color="red">9</font> | ||
|- | |- | ||
| − | | | + | | || || || || || |
| − | | <font color="red"> | + | | <font color="red">10</font> |
| − | | || · | + | | || · || || 1 || || 1 || || 36|| || 1 || || 2 || || 1 || || 8 || || · || |
| − | | <font color="red"> | + | | <font color="red">10</font> |
|- | |- | ||
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| − | | | + | | || · || || 1 || || 1 || || 49 || || 1 || || 5 || || 1 || || 27 || || 1 || || · || |
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| − | | | + | |- |
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| − | | | + | | || · || || 10 || || 3 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || · || |
| − | | | + | | <font color="red">12</font> |
| − | | | + | |- |
| − | | | + | | || || |
| − | | | + | | <font color="red">13</font> |
| − | | width="3%" | | + | | || · || || 11 || || 1 || || 1 || || 2 || || 7 || || 1 || || 125 || || 4 || || 3 || || 1 || || · || |
| + | | <font color="red">13</font> | ||
| + | |- | ||
| + | | || | ||
| + | | <font color="red">14</font> | ||
| + | | || · || || 3 || || 1 || || 100 || || 1 || || 1 || || 1 || || 216 || || 1 || || 1 || || 1 || || 4 || || · || | ||
| + | | <font color="red">14</font> | ||
| + | |- | ||
| + | | | ||
| + | | <font color="red">15</font> | ||
| + | | || · || || 13 || || 2 || || 121 || || 1 || || 3 || || 1 || || 343 || || 1 || || 5 || || 1 || || 9 || || 1 || || · || | ||
| + | | <font color="red">15</font> | ||
| + | |- | ||
| + | | width="3%" | <font color="red">16</font> | ||
| width="3%" | | | width="3%" | | ||
| − | | width="3%" | | + | | width="3%" | · |
| width="3%" | | | width="3%" | | ||
| − | | width="3%" | | + | | width="3%" | 14 |
| width="3%" | | | width="3%" | | ||
| width="3%" | 1 | | width="3%" | 1 | ||
| width="3%" | | | width="3%" | | ||
| − | | width="3%" | | + | | width="3%" | 9 |
| width="3%" | | | width="3%" | | ||
| width="3%" | 1 | | width="3%" | 1 | ||
| width="3%" | | | width="3%" | | ||
| − | | width="3%" | 2 | + | | width="3%" | 10 |
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 6 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 1 | ||
| + | | width="3%" | | ||
| + | | width="3%" | 2 | ||
| width="3%" | | | width="3%" | | ||
| width="3%" | 1 | | width="3%" | 1 | ||
| Line 2,271: | Line 2,504: | ||
* [http://oeis.org/wiki/A109301 A109301] | * [http://oeis.org/wiki/A109301 A109301] | ||
| + | |||
| + | ===Example=== | ||
| + | |||
| + | : <math>802701 = 9 \cdot 89189 = \text{p}_2^2 \text{p}_{8638}^1</math> | ||
| + | |||
| + | : <math>\text{Writing}~ (\operatorname{prime}(i))^j ~\text{as}~ i\!:\!j, ~\text{we have:}</math> | ||
| + | |||
| + | : <math>\begin{array}{lllll} | ||
| + | 802701 | ||
| + | & = & 9 \cdot 89189 | ||
| + | & = & 2\!:\!2 ~~ 8638\!:\!1 | ||
| + | \\ | ||
| + | 8638 | ||
| + | & = & 2 \cdot 7 \cdot 617 | ||
| + | & = & 1\!:\!1 ~~ 4\!:\!1 ~~ 113\!:\!1 | ||
| + | \\ | ||
| + | 113 | ||
| + | & & | ||
| + | & = & 30\!:\!1 | ||
| + | \\ | ||
| + | 30 | ||
| + | & = & 2 \cdot 3 \cdot 5 | ||
| + | & = & 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 | ||
| + | \\ | ||
| + | 4 | ||
| + | & & | ||
| + | & = & 1\!:\!2 | ||
| + | \\ | ||
| + | 3 | ||
| + | & & | ||
| + | & = & 2\!:\!1 | ||
| + | \\ | ||
| + | 2 | ||
| + | & & | ||
| + | & = & 1\!:\!1 | ||
| + | \end{array}</math> | ||
| + | |||
| + | : <math>\text{So the rote of 802701 is the following graph:}\!</math> | ||
| + | |||
| + | :{| border="1" cellpadding="20" | ||
| + | | [[Image:Rote 802701 Big.jpg|330px]] | ||
| + | |} | ||
| + | |||
| + | : <math>\text{By inspection, the rote height of 802701 is 6.}\!</math> | ||
===JPEG=== | ===JPEG=== | ||
| Line 2,276: | Line 2,553: | ||
{| align="center" border="1" cellpadding="6" | {| align="center" border="1" cellpadding="6" | ||
| valign="bottom" | | | valign="bottom" | | ||
| − | <p>[[Image: | + | <p>[[Image:Rote 1 Big.jpg|20px]]</p><br> |
| − | <p><math>\ | + | <p><math>1\!</math></p><br> |
| + | <p><math>a(1) ~=~ 0</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 2 Big.jpg|40px]]</p><br> | <p>[[Image:Rote 2 Big.jpg|40px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}\!</math></p><br> |
| + | <p><math>a(2) ~=~ 1</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 3 Big.jpg|40px]]</p><br> | <p>[[Image:Rote 3 Big.jpg|40px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_\text{p}\!</math></p><br> |
| + | <p><math>a(3) ~=~ 2</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 4 Big.jpg|65px]]</p><br> | <p>[[Image:Rote 4 Big.jpg|65px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}^\text{p}\!</math></p><br> |
| + | <p><math>a(4) ~=~ 2</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 5 Big.jpg|40px]]</p><br> | <p>[[Image:Rote 5 Big.jpg|40px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(5) ~=~ 3</math></p> | ||
|- | |- | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 6 Big.jpg|80px]]</p><br> | <p>[[Image:Rote 6 Big.jpg|80px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p} \text{p}_\text{p}\!</math></p><br> |
| + | <p><math>a(6) ~=~ 2</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 7 Big.jpg|65px]]</p><br> | <p>[[Image:Rote 7 Big.jpg|65px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(7) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 8 Big.jpg|65px]]</p><br> | <p>[[Image:Rote 8 Big.jpg|65px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}^{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(8) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 9 Big.jpg|80px]]</p><br> | <p>[[Image:Rote 9 Big.jpg|80px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_\text{p}^\text{p}\!</math></p><br> |
| + | <p><math>a(9) ~=~ 2</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 10 Big.jpg|80px]]</p><br> | <p>[[Image:Rote 10 Big.jpg|80px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(10) ~=~ 3</math></p> | ||
|- | |- | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 11 Big.jpg|40px]]</p><br> | <p>[[Image:Rote 11 Big.jpg|40px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(11) ~=~ 4</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 12 Big.jpg|105px]]</p><br> | <p>[[Image:Rote 12 Big.jpg|105px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}^\text{p} \text{p}_\text{p}\!</math></p><br> |
| + | <p><math>a(12) ~=~ 2</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 13 Big.jpg|80px]]</p><br> | <p>[[Image:Rote 13 Big.jpg|80px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(13) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 14 Big.jpg|105px]]</p><br> | <p>[[Image:Rote 14 Big.jpg|105px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(14) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 15 Big.jpg|80px]]</p><br> | <p>[[Image:Rote 15 Big.jpg|80px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(15) ~=~ 3</math></p> | ||
|- | |- | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 16 Big.jpg|90px]]</p><br> | <p>[[Image:Rote 16 Big.jpg|90px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}^{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(16) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 17 Big.jpg|65px]]</p><br> | <p>[[Image:Rote 17 Big.jpg|65px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br> |
| + | <p><math>a(17) ~=~ 4</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 18 Big.jpg|120px]]</p><br> | <p>[[Image:Rote 18 Big.jpg|120px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br> |
| + | <p><math>a(18) ~=~ 2</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 19 Big.jpg|65px]]</p><br> | <p>[[Image:Rote 19 Big.jpg|65px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(19) ~=~ 4</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 20 Big.jpg|105px]]</p><br> | <p>[[Image:Rote 20 Big.jpg|105px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(20) ~=~ 3</math></p> | ||
|- | |- | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 21 Big.jpg|105px]]</p><br> | <p>[[Image:Rote 21 Big.jpg|105px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(21) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 22 Big.jpg|80px]]</p><br> | <p>[[Image:Rote 22 Big.jpg|80px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(22) ~=~ 4</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 23 Big.jpg|80px]]</p><br> | <p>[[Image:Rote 23 Big.jpg|80px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(23) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 24 Big.jpg|105px]]</p><br> | <p>[[Image:Rote 24 Big.jpg|105px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!</math></p><br> |
| + | <p><math>a(24) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 25 Big.jpg|80px]]</p><br> | <p>[[Image:Rote 25 Big.jpg|80px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br> |
| + | <p><math>a(25) ~=~ 3</math></p> | ||
|- | |- | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 26 Big.jpg|120px]]</p><br> | <p>[[Image:Rote 26 Big.jpg|120px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(26) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 27 Big.jpg|80px]]</p><br> | <p>[[Image:Rote 27 Big.jpg|80px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(27) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 28 Big.jpg|130px]]</p><br> | <p>[[Image:Rote 28 Big.jpg|130px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(28) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 29 Big.jpg|80px]]</p><br> | <p>[[Image:Rote 29 Big.jpg|80px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(29) ~=~ 4</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 30 Big.jpg|120px]]</p><br> | <p>[[Image:Rote 30 Big.jpg|120px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(30) ~=~ 3</math></p> | ||
|- | |- | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 31 Big.jpg|40px]]</p><br> | <p>[[Image:Rote 31 Big.jpg|40px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!</math></p><br> |
| + | <p><math>a(31) ~=~ 5</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 32 Big.jpg|65px]]</p><br> | <p>[[Image:Rote 32 Big.jpg|65px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}^{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(32) ~=~ 4</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 33 Big.jpg|80px]]</p><br> | <p>[[Image:Rote 33 Big.jpg|80px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(33) ~=~ 4</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 34 Big.jpg|105px]]</p><br> | <p>[[Image:Rote 34 Big.jpg|105px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br> |
| + | <p><math>a(34) ~=~ 4</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 35 Big.jpg|105px]]</p><br> | <p>[[Image:Rote 35 Big.jpg|105px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(35) ~=~ 3</math></p> | ||
|- | |- | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 36 Big.jpg|145px]]</p><br> | <p>[[Image:Rote 36 Big.jpg|145px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br> |
| + | <p><math>a(36) ~=~ 2</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 37 Big.jpg|105px]]</p><br> | <p>[[Image:Rote 37 Big.jpg|105px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(37) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 38 Big.jpg|105px]]</p><br> | <p>[[Image:Rote 38 Big.jpg|105px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(38) ~=~ 4</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 39 Big.jpg|120px]]</p><br> | <p>[[Image:Rote 39 Big.jpg|120px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(39) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 40 Big.jpg|105px]]</p><br> | <p>[[Image:Rote 40 Big.jpg|105px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(40) ~=~ 3</math></p> | ||
|- | |- | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 41 Big.jpg|80px]]</p><br> | <p>[[Image:Rote 41 Big.jpg|80px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(41) ~=~ 4</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 42 Big.jpg|145px]]</p><br> | <p>[[Image:Rote 42 Big.jpg|145px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(42) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 43 Big.jpg|105px]]</p><br> | <p>[[Image:Rote 43 Big.jpg|105px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!</math></p><br> |
| + | <p><math>a(43) ~=~ 4</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 44 Big.jpg|105px]]</p><br> | <p>[[Image:Rote 44 Big.jpg|105px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(44) ~=~ 4</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 45 Big.jpg|120px]]</p><br> | <p>[[Image:Rote 45 Big.jpg|120px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(45) ~=~ 3</math></p> | ||
|- | |- | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 46 Big.jpg|120px]]</p><br> | <p>[[Image:Rote 46 Big.jpg|120px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(46) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 47 Big.jpg|80px]]</p><br> | <p>[[Image:Rote 47 Big.jpg|80px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(47) ~=~ 4</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 48 Big.jpg|105px]]</p><br> | <p>[[Image:Rote 48 Big.jpg|105px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!</math></p><br> |
| + | <p><math>a(48) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 49 Big.jpg|80px]]</p><br> | <p>[[Image:Rote 49 Big.jpg|80px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p}^\text{p}}^\text{p}\!</math></p><br> |
| + | <p><math>a(49) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 50 Big.jpg|120px]]</p><br> | <p>[[Image:Rote 50 Big.jpg|120px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br> |
| + | <p><math>a(50) ~=~ 3</math></p> | ||
|- | |- | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 51 Big.jpg|105px]]</p><br> | <p>[[Image:Rote 51 Big.jpg|105px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br> |
| + | <p><math>a(51) ~=~ 4</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 52 Big.jpg|145px]]</p><br> | <p>[[Image:Rote 52 Big.jpg|145px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(52) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 53 Big.jpg|90px]]</p><br> | <p>[[Image:Rote 53 Big.jpg|90px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p}^{\text{p}^\text{p}}}\!</math></p><br> |
| + | <p><math>a(53) ~=~ 4</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 54 Big.jpg|120px]]</p><br> | <p>[[Image:Rote 54 Big.jpg|120px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(54) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 55 Big.jpg|80px]]</p><br> | <p>[[Image:Rote 55 Big.jpg|80px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(55) ~=~ 4</math></p> | ||
|- | |- | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 56 Big.jpg|130px]]</p><br> | <p>[[Image:Rote 56 Big.jpg|130px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(56) ~=~ 3</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 57 Big.jpg|105px]]</p><br> | <p>[[Image:Rote 57 Big.jpg|105px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(57) ~=~ 4</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 58 Big.jpg|120px]]</p><br> | <p>[[Image:Rote 58 Big.jpg|120px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(58) ~=~ 4</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 59 Big.jpg|65px]]</p><br> | <p>[[Image:Rote 59 Big.jpg|65px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!</math></p><br> |
| + | <p><math>a(59) ~=~ 5</math></p> | ||
| valign="bottom" | | | valign="bottom" | | ||
<p>[[Image:Rote 60 Big.jpg|155px]]</p><br> | <p>[[Image:Rote 60 Big.jpg|155px]]</p><br> | ||
| − | <p><math>\ | + | <p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(60) ~=~ 3</math></p> | ||
|} | |} | ||
| Line 2,614: | Line 2,951: | ||
* while g(3) = 1:36 2:36 3:36 4:36 6:36 9:36 12:36 18:36 36:36 = | * while g(3) = 1:36 2:36 3:36 4:36 6:36 9:36 12:36 18:36 36:36 = | ||
* (2 3 5 7 13 23 37 61 151)^36 = 21399271530^36 = roughly | * (2 3 5 7 13 23 37 61 151)^36 = 21399271530^36 = roughly | ||
| − | * 7.840858554516122655953405327738 x 10^371. | + | * 7.840858554516122655953405327738 x 10^371. |
| + | |||
| + | Example | ||
| + | |||
| + | * Writing (prime(i))^j as i:j, we have: | ||
| + | * 802701 = 2:2 8638:1 | ||
| + | * 8638 = 1:1 4:1 113:1 | ||
| + | * 113 = 30:1 | ||
| + | * 30 = 1:1 2:1 3:1 | ||
| + | * 4 = 1:2 | ||
| + | * 3 = 2:1 | ||
| + | * 2 = 1:1 | ||
| + | * 1 = { } | ||
| + | * So rote(802701) is the graph: | ||
| + | * | ||
| + | * o-o | ||
| + | * | | ||
| + | * o-o o-o | ||
| + | * | | | ||
| + | * o-o o-o o-o o-o | ||
| + | * | | | | | ||
| + | * o-o o===o===o-o | ||
| + | * | | | ||
| + | * o-o o-o o-o o-o o---------o | ||
| + | * | | | | | | ||
| + | * o---o o===o=====o---------o | ||
| + | * | | | ||
| + | * O=======O | ||
| + | * | ||
| + | * Therefore rhig(802701) = 6. | ||
</pre> | </pre> | ||
Latest revision as of 18:48, 31 January 2010
A061396
Plain Wiki Table
Large Scale
| \(\text{Integer}\!\) | \(\text{Factorization}\!\) | \(\text{Notation}\!\) | \(\text{Riff Digraph}\!\) | \(\text{Rote Graph}\!\) | \(\text{Traversal}\!\) |
| \(1\!\) | \(1\!\) | 
|
|||
| \(2\!\) | \(\text{p}_1^1\!\) | \(\text{p}\!\) | 
|
|
\(((~))\) |
| \(3\!\) |
\(\begin{array}{lll} \text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1 \end{array}\) |
\(\text{p}_\text{p}\!\) | 
|
|
\((((~))(~))\) |
| \(4\!\) |
\(\begin{array}{lll} \text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1} \end{array}\) |
\(\text{p}^\text{p}\!\) | 
|
|
\(((((~))))\) |
| \(5\!\) |
\(\begin{array}{lll}
\text{p}_3^1
& = & \text{p}_{\text{p}_2^1}^1
\'"`UNIQ-MathJax1-QINU`"'
'"`UNIQ-MathJax2-QINU`"'
'"`UNIQ-MathJax3-QINU`"'
'"`UNIQ-MathJax4-QINU`"'
:{| border="1" cellpadding="20"
| [[Image:Rote 802701 Big.jpg|330px]]
|}
'"`UNIQ-MathJax5-QINU`"'
==='"`UNIQ--h-40--QINU`"'JPEG===
{| align="center" border="1" cellpadding="6"
| valign="bottom" |
<p>[[Image:Rote 1 Big.jpg|20px]]</p><br>
<p>\(1\!\) \(a(1) ~=~ 0\) |
\(\text{p}\!\) \(a(2) ~=~ 1\) |
\(\text{p}_\text{p}\!\) \(a(3) ~=~ 2\) |
\(\text{p}^\text{p}\!\) \(a(4) ~=~ 2\) |
\(\text{p}_{\text{p}_\text{p}}\!\) \(a(5) ~=~ 3\) |
|
\(\text{p} \text{p}_\text{p}\!\) \(a(6) ~=~ 2\) |
\(\text{p}_{\text{p}^\text{p}}\!\) \(a(7) ~=~ 3\) |
\(\text{p}^{\text{p}_\text{p}}\!\) \(a(8) ~=~ 3\) |
\(\text{p}_\text{p}^\text{p}\!\) \(a(9) ~=~ 2\) |
\(\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(10) ~=~ 3\) | |
|
\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(11) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_\text{p}\!\) \(a(12) ~=~ 2\) |
\(\text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(13) ~=~ 3\) |
\(\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(14) ~=~ 3\) |
\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(15) ~=~ 3\) | |
|
\(\text{p}^{\text{p}^\text{p}}\!\) \(a(16) ~=~ 3\) |
\(\text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(17) ~=~ 4\) |
\(\text{p} \text{p}_\text{p}^\text{p}\!\) \(a(18) ~=~ 2\) |
\(\text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(19) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(20) ~=~ 3\) | |
|
\(\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(21) ~=~ 3\) |
\(\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(22) ~=~ 4\) |
\(\text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(a(23) ~=~ 3\) |
\(\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!\) \(a(24) ~=~ 3\) |
\(\text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(a(25) ~=~ 3\) | |
|
\(\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(26) ~=~ 3\) |
\(\text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(a(27) ~=~ 3\) |
\(\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(28) ~=~ 3\) |
\(\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(29) ~=~ 4\) |
\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(30) ~=~ 3\) | |
|
\(\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!\) \(a(31) ~=~ 5\) |
\(\text{p}^{\text{p}_{\text{p}_\text{p}}}\!\) \(a(32) ~=~ 4\) |
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(33) ~=~ 4\) |
\(\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(34) ~=~ 4\) |
\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(a(35) ~=~ 3\) | |
|
\(\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!\) \(a(36) ~=~ 2\) |
\(\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!\) \(a(37) ~=~ 3\) |
\(\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(38) ~=~ 4\) |
\(\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(39) ~=~ 3\) |
\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!\) \(a(40) ~=~ 3\) | |
|
\(\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!\) \(a(41) ~=~ 4\) |
\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(a(42) ~=~ 3\) |
\(\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!\) \(a(43) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(44) ~=~ 4\) |
\(\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(45) ~=~ 3\) | |
|
\(\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(a(46) ~=~ 3\) |
\(\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(47) ~=~ 4\) |
\(\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!\) \(a(48) ~=~ 3\) |
\(\text{p}_{\text{p}^\text{p}}^\text{p}\!\) \(a(49) ~=~ 3\) |
\(\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(a(50) ~=~ 3\) | |
|
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(a(51) ~=~ 4\) |
\(\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(a(52) ~=~ 3\) |
\(\text{p}_{\text{p}^{\text{p}^\text{p}}}\!\) \(a(53) ~=~ 4\) |
\(\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(a(54) ~=~ 3\) |
\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(a(55) ~=~ 4\) | |
|
\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(a(56) ~=~ 3\) |
\(\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(a(57) ~=~ 4\) |
\(\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(a(58) ~=~ 4\) |
\(\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!\) \(a(59) ~=~ 5\) |
\(\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(a(60) ~=~ 3\) |
ASCII
Comment
* Table of Rotes and Primal Functions for Positive Integers from 1 to 40
*
* o-o
* |
* o-o o-o o-o
* | | |
* o-o o-o o-o o-o
* | | | |
* O O O O O
*
* { } 1:1 2:1 1:2 3:1
*
* 1 2 3 4 5
*
*
* o-o o-o o-o
* | | |
* o-o o-o o-o o-o o-o o-o
* | | | | | |
* o-o o-o o-o o-o o---o o-o o-o
* | | | | | | |
* O===O O O O O===O
*
* 1:1 2:1 4:1 1:3 2:2 1:1 3:1
*
* 6 7 8 9 10
*
*
* o-o
* |
* o-o o-o o-o o-o
* | | | |
* o-o o-o o-o o-o o-o o-o o-o o-o
* | | | | | | | |
* o-o o-o o-o o===o-o o-o o-o o-o o-o
* | | | | | | | |
* O O=====O O O===O O===O
*
* 5:1 1:2 2:1 6:1 1:1 4:1 2:1 3:1
*
* 11 12 13 14 15
*
*
* o-o o-o
* | |
* o-o o-o o-o o-o
* | | | |
* o-o o-o o-o o-o o-o o-o o-o
* | | | | | | |
* o-o o-o o-o o---o o-o o-o o-o
* | | | | | | |
* O O O===O O O=====O
*
* 1:4 7:1 1:1 2:2 8:1 1:2 3:1
*
* 16 17 18 19 20
*
*
* o-o
* |
* o-o o-o o-o o-o o-o o-o
* | | | | | |
* o-o o-o o-o o---o o-o o-o o-o o-o
* | | | | | | | |
* o-o o-o o-o o-o o-o o-o o-o o---o
* | | | | | | | |
* O===O O===O O O=====O O
*
* 2:1 4:1 1:1 5:1 9:1 1:3 2:1 3:2
*
* 21 22 23 24 25
*
*
* o-o
* |
* o-o o-o o-o o-o o-o
* | | | | |
* o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o
* | | | | | | | | | |
* o-o o===o-o o---o o-o o-o o===o-o o-o o-o o-o
* | | | | | | | | |
* O===O O O=====O O O===O===O
*
* 1:1 6:1 2:3 1:2 4:1 10:1 1:1 2:1 3:1
*
* 26 27 28 29 30
*
*
* o-o
* |
* o-o o-o o-o o-o
* | | | |
* o-o o-o o-o o-o o-o o-o
* | | | | | |
* o-o o-o o-o o-o o-o o-o o-o
* | | | | | | |
* o-o o-o o-o o-o o-o o-o o-o o-o
* | | | | | | | |
* O O O===O O===O O===O
*
* 11:1 1:5 2:1 5:1 1:1 7:1 3:1 4:1
*
* 31 32 33 34 35
*
*
* o-o
* |
* o-o o-o o-o o-o o-o o-o
* | | | | | |
* o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o
* | | | | | | | | | | |
* o-o o---o o=====o-o o-o o-o o-o o===o-o o-o o-o
* | | | | | | | | |
* O=====O O O===O O===O O=====O
*
* 1:2 2:2 12:1 1:1 8:1 2:1 6:1 1:3 3:1
*
* 36 37 38 39 40
*
* In these Figures, "extended lines of identity" like o===o
* indicate identified nodes and capital O is the root node.
* The rote height in gammas is found by finding the number
* of graphs of the following shape between the root and one
* of the highest nodes of the tree:
* o--o
* |
* o
* A sequence like this, that can be regarded as a nonnegative integer
* measure on positive integers, may have as many as 3 other sequences
* associated with it. Given that the fiber of a function f at n is all
* the domain elements that map to n, we always have the fiber minimum
* or minimum inverse function and may also have the fiber cardinality
* and the fiber maximum or maximum inverse function. For A109301, the
* minimum inverse is A007097(n) = min {k : A109301(k) = n}, giving the
* first positive integer whose rote height is n, the fiber cardinality
* is A109300, giving the number of positive integers of rote height n,
* while the maximum inverse, g(n) = max {k : A109301(k) = n}, giving
* the last positive integer whose rote height is n, has the following
* initial terms: g(0) = { } = 1, g(1) = 1:1 = 2, g(2) = 1:2 2:2 = 36,
* while g(3) = 1:36 2:36 3:36 4:36 6:36 9:36 12:36 18:36 36:36 =
* (2 3 5 7 13 23 37 61 151)^36 = 21399271530^36 = roughly
* 7.840858554516122655953405327738 x 10^371.
Example
* Writing (prime(i))^j as i:j, we have:
* 802701 = 2:2 8638:1
* 8638 = 1:1 4:1 113:1
* 113 = 30:1
* 30 = 1:1 2:1 3:1
* 4 = 1:2
* 3 = 2:1
* 2 = 1:1
* 1 = { }
* So rote(802701) is the graph:
*
* o-o
* |
* o-o o-o
* | |
* o-o o-o o-o o-o
* | | | |
* o-o o===o===o-o
* | |
* o-o o-o o-o o-o o---------o
* | | | | |
* o---o o===o=====o---------o
* | |
* O=======O
*
* Therefore rhig(802701) = 6.
A111795
JPEG
|
\(\begin{array}{l} \varnothing \\ 1 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 \\ 2 \end{array}\) |
\(\begin{array}{l} 2\!:\!1 \\ 3 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 \\ 4 \end{array}\) |
\(\begin{array}{l} 3\!:\!1 \\ 5 \end{array}\) |
|
\(\begin{array}{l} 4\!:\!1 \\ 7 \end{array}\) |
\(\begin{array}{l} 1\!:\!3 \\ 8 \end{array}\) |
\(\begin{array}{l} 5\!:\!1 \\ 11 \end{array}\) |
\(\begin{array}{l} 1\!:\!4 \\ 16 \end{array}\) |
\(\begin{array}{l} 7\!:\!1 \\ 17 \end{array}\) |
|
\(\begin{array}{l} 8\!:\!1 \\ 19 \end{array}\) |
\(\begin{array}{l} 11\!:\!1 \\ 31 \end{array}\) |
\(\begin{array}{l} 1\!:\!5 \\ 32 \end{array}\) |
\(\begin{array}{l} 16\!:\!1 \\ 53 \end{array}\) |
\(\begin{array}{l} 17\!:\!1 \\ 59 \end{array}\) |
ASCII
Example
* Tables of Rotes and Primal Codes for a(1) to a(9)
*
* o-o
* |
* o-o o-o o-o o-o o-o
* | | | | |
* o-o o-o o-o o-o o-o o-o o-o
* | | | | | | |
* o-o o-o o-o o-o o-o o-o o-o o-o
* | | | | | | | |
* O O O O O O O O O
*
* { } 1:1 2:1 1:2 3:1 4:1 1:3 5:1 1:4
*
* 1 2 3 4 5 7 8 11 16
*
A111800
TeX + JPEG
\(\text{Writing}~ \operatorname{prime}(i)^j ~\text{as}~ i\!:\!j, 2500 = 4 \cdot 625 = 2^2 5^4 = 1\!:\!2 ~~ 3\!:\!4 ~\text{has the following rote:}\)
|
\(\text{So}~ a(2500) = a(1\!:\!2 ~~ 3\!:\!4) = a(1) + a(2) + a(3) + a(4) + 1 = 1 + 3 + 5 + 5 + 1 = 15.\)
ASCII
Example
* Writing prime(i)^j as i:j and using equal signs between identified nodes:
* 2500 = 4 * 625 = 2^2 5^4 = 1:2 3:4 has the following rote:
*
* o-o o-o
* | |
* o-o o-o o-o
* | | |
* o-o o---o
* | |
* O=====O
*
* So a(2500) = a(1:2 3:4) = a(1)+a(2)+a(3)+a(4)+1 = 1+3+5+5+1 = 15.
