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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A062860 Smallest j with n nodes in its riff (rooted index-functional forest).

Original entry on oeis.org

1, 2, 3, 5, 10, 15, 30, 55, 105, 165, 330, 660, 1155, 2145, 4290, 7755, 15015, 30030, 54285, 100815, 201630, 403260, 705705, 1411410, 2822820, 5645640, 11392095, 20465445, 40930890, 79744665, 159489330, 318978660, 637957320, 1321483020
Offset: 0

Views

Author

Jon Awbrey, Jun 25 2001

Keywords

Comments

Greatest j with n nodes in its riff is given by A014221(n+1).

Links

Crossrefs

Cf. A014221, A061396, A062504, A062537, A111800.

Extensions

More terms from David W. Wilson, Jun 26 2001

A109301 a(n) = rhig(n) = rote height in gammas of n, where the "rote" corresponding to a positive integer n is a graph derived from the primes factorization of n, as illustrated in the comments.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 3, 3, 2, 3, 4, 2, 3, 3, 3, 3, 4, 2, 4, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 5, 4, 4, 4, 3, 2, 3, 4, 3, 3, 4, 3, 4, 4, 3, 3, 4, 3, 3, 3, 4, 3, 4, 3, 4, 3, 4, 4, 5, 3, 3, 5, 3, 3, 3, 4, 5, 4, 3, 3, 4, 3, 4, 3, 3, 4, 4, 3, 5, 3, 3, 4, 4, 3, 4, 4, 4, 4, 4, 3, 3, 3, 5, 4, 4, 4, 4, 3, 4, 3
Offset: 1

Views

Author

Jon Awbrey, Jun 24 2005 - Jul 08 2005

Keywords

Comments

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, which 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; 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.

Examples

			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.

Links

Crossrefs

Cf. A007097, A050924, A061396, A062504, A062537, A062860.
Cf. A106177, A108352, A108371, A109300, A111791 to A111800.

Formula

Writing (prime(i))^j as i:j, the prime factorization of a positive integer n can be written as n = prod_(k = 1 to m) i(k):j(k). This sets up the formula: rhig(n) = 1 + max_(k = 1 to m) {rhig(i(k)), rhig(j(k))}, where rhig(1) = 0.

A109300 a(n) = number of positive integers whose rote height in gammas is n.

Original entry on oeis.org

1, 1, 7, 999999991
Offset: 0

Views

Author

Jon Awbrey, Jul 04 2005, revised Sep 06 2005

Keywords

Comments

a(n) is the sequence of first differences of A050924. Conversely, A050924 is the sequence of partial sums of a(n). This can be seen as follows. Let P(0) c P(1) c ... c P(n) c ... be an increasing sequence of sets of partial functions that is defined by the recursion: P(0) = {the empty function}, P(n+1) = {partial functions: P(n) -> P(n)}. Then |P(n)| = A050924(n+1) = number of positive integers of rote height at most n, hence |P(n)| - |P(n-1)| = a(n) = number of positive integers of rote height exactly n.

Examples

			Table of Rotes and Primal Functions for Positive Integers of Rote Height 2
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
o-o ` ` o-o ` ` ` o-o ` o-o o-o ` ` o-o o-o ` ` ` o-o o-o ` ` o-o o-o o-o
| ` ` ` | ` ` ` ` | ` ` | ` | ` ` ` | ` | ` ` ` ` | ` | ` ` ` | ` | ` | `
o-o ` o-o ` ` o-o o-o ` o---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 ` 1:2 ` ` 1:1 2:1 ` 2:2 ` ` ` 1:2 2:1 ` ` 1:1 2:2 ` ` ` 1:2 2:2 ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
3 ` ` 4 ` ` ` 6 ` ` ` ` 9 ` ` ` ` 12` ` ` ` ` 18` ` ` ` ` ` 36` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Links

Crossrefs

Cf. A007097, A050924, A061396, A062504, A062537, A062860.
Cf. A109301, A106177, A108352, A108371, A111791 to A111800.

Formula

a(n) is defined by the recursion a(n+1) = (b(n) + 1)^b(n) - b(n), where a(0) = 1 and b(n) = Sum_[0, n] a(i).

A111791 Positive integers sorted by rote height, as measured by A109301.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 18, 36, 5, 7, 8, 10, 13, 14, 15, 16, 20, 21, 23, 24, 25, 26, 27, 28, 30, 35, 37, 39, 40, 42, 45, 46, 48, 49, 50, 52, 54, 56, 60, 61, 63, 64, 65, 69, 70, 72, 74, 75, 78, 80, 81, 84, 90, 91, 92, 98, 100
Offset: 1

Views

Author

Jon Awbrey, Aug 24 2005, revised Sep 02 2005

Keywords

Examples

			Table in which the h-th row lists the positive integers of rote height h:
h | m such that rhig(m) = A109301(m) = h
--+------------------------------------------------------
0 |  1
--+------------------------------------------------------
1 |  2
--+------------------------------------------------------
2 |  3  4  6  9 12 18 36
--+------------------------------------------------------
3 |  5  7  8 10 13 14 15 16 20 21 23 24 25 26 27  28 30
  | 35 37 39 40 42 45 46 48 49 50 52 54 56 60 61  63
  | 64 65 69 70 72 74 75 78 80 81 84 90 91 92 98 100 ...
--+------------------------------------------------------
4 | 11 17 19 22 29 32 33 34 38 41 43 44 47 51 53 55
  | 57 58 66 68 71 73 76 77 82 83 85 86 87 88 89 94
  | 95 96 97 99 ...
--+------------------------------------------------------
5 | 31 59 62 67 79 93 ...
--+------------------------------------------------------
First column = A007097. Count in h^th row = A109300(h).
Cumulative count up through the h^th row = A050924(h+1).

Links

Crossrefs

Cf. A007097, A050924, A061396, A062504, A062537, A062860.
Cf. A109300, A109301, A111792 to A111800.

A111793 Triangle T(g, h) = number of rotes of weight g and height h, both in gammas.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 10, 8, 1, 24, 32, 16
Offset: 1

Views

Author

Jon Awbrey, Aug 26 2005, revised Aug 28 2005

Keywords

Comments

T(g, h) = |{positive integers m : A062537(m) = g and A109301(m) = h}|.
Row sums = A061396. Column sums = A109300. See A111792 for details.
Main diagonal T(j, j) = 2^(j-1) for j > 0, T(0, 0) = 1.

Examples

			Table T(g, h), omitting zeros, starts out as follows:
g\h| 0 ` 1 ` 2 ` 3 ` 4 ` 5
---+-----------------------
`0 | 1
`1 | ` ` 1
`2 | ` ` ` ` 2
`3 | ` ` ` ` 2 ` 4
`4 | ` ` ` ` 2 `10 ` 8
`5 | ` ` ` ` 1 `24 `32 `16

Links

Crossrefs

Cf. A007097, A050924, A061396, A062504, A062537, A062860.
Cf. A109300, A109301, A111791, A111792, A111794, A111800.

A112846 Number of riffs on n or fewer nodes. Number of rotes on 2n+1 or fewer nodes.

Original entry on oeis.org

1, 2, 4, 10, 30, 103, 384, 1508, 6126, 25513, 108278, 466523, 2034981, 8968746, 39875940, 138760603, 178636543, 3026583484, 16028356176, 75647274620, 350111055991, 1618175863400, 7495933933620, 34821723061950
Offset: 0

Views

Author

Jon Awbrey, Oct 04 2005, based on calculations by Vladeta Jovovic and David W. Wilson

Keywords

Comments

Partial sums of A061396.

Links

Crossrefs

Cf. A000079, A001221, A007097, A014221, A050924.
Cf. A061396, A062504, A062537, A062860, A106177.
Cf. A109300, A109301, A111791 to A111800.
Cf. A112095, A112096, A112480, A112481.

A111792 Positive integers sorted by rote weight (A062537) and rote height (A109301).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 5, 7, 8, 16, 12, 18, 10, 13, 14, 23, 25, 27, 49, 64, 81, 512, 11, 17, 19, 32, 53, 128, 256, 65536
Offset: 1

Views

Author

Jon Awbrey, Aug 25 2005, revised Aug 27 2005

Keywords

Examples

			Table of Integers, Primal Codes, Sort Parameters and Subtotals
` ` a ` code` ` | g h | s | t
----------------+-----+---+---
` ` 1 = { } ` ` | 0 0 | 1 | 1
----------------+-----+---+---
` ` 2 = 1:1 ` ` | 1 1 | 1 | 1
----------------+-----+---+---
` ` 3 = 2:1 ` ` | 2 2 | ` |
` ` 4 = 1:2 ` ` | 2 2 | 2 | 2
----------------+-----+---+---
` ` 6 = 1:1 2:1 | 3 2 | ` |
` ` 9 = 2:2 ` ` | 3 2 | 2 |
----------------+-----+---+---
` ` 5 = 3:1 ` ` | 3 3 | ` |
` ` 7 = 4:1 ` ` | 3 3 | ` |
` ` 8 = 1:3 ` ` | 3 3 | ` |
` `16 = 1:4 ` ` | 3 3 | 4 | 6
----------------+-----+---+---
` `12 = 1:2 2:1 | 4 2 | ` |
` `18 = 1:1 2:2 | 4 2 | 2 |
----------------+-----+---+---
` `10 = 1:1 3:1 | 4 3 | ` |
` `13 = 6:1 ` ` | 4 3 | ` |
` `14 = 1:1 4:1 | 4 3 | ` |
` `23 = 9:1 ` ` | 4 3 | ` |
` `25 = 3:2 ` ` | 4 3 | ` |
` `27 = 2:3 ` ` | 4 3 | ` |
` `49 = 4:2 ` ` | 4 3 | ` |
` `64 = 1:6 ` ` | 4 3 | ` |
` `81 = 2:4 ` ` | 4 3 | ` |
` 512 = 1:9 ` ` | 4 3 |10 |
----------------+-----+---+---
` `11 = 5:1 ` ` | 4 4 | ` |
` `17 = 7:1 ` ` | 4 4 | ` |
` `19 = 8:1 ` ` | 4 4 | ` |
` `32 = 1:5 ` ` | 4 4 | ` |
` `53 = 16:1` ` | 4 4 | ` |
` 128 = 1:7 ` ` | 4 4 | ` |
` 256 = 1:8 ` ` | 4 4 | ` |
65536 = 1:16` ` | 4 4 | 8 |20
----------------+-----+---+---
a = this sequence
g = rote weight in gammas = A062537
h = rote height in gammas = A109301
s = count in (g, h) class = A111793
t = count in weight class = A061396

Links

Crossrefs

Cf. A007097, A050924, A061396, A062504, A062537, A062860.
Cf. A109300, A109301, A111791, A111793, A111800.

A112095 Positive integers sorted by rote weight, rote height and rote wayage.

Original entry on oeis.org

1, 2, 3, 4, 9, 6, 5, 7, 8, 16, 12, 18, 13, 23, 25, 27, 49, 64, 81, 512, 10, 14, 11, 17, 19, 32, 53, 128, 256, 65536, 36, 37, 61, 125, 169, 343, 529, 625, 729, 2401, 4096, 19683, 262144, 15, 20, 21, 24, 26, 28, 46, 48, 50, 54, 98, 162, 29, 41, 43, 83, 97, 103, 121, 227
Offset: 1

Views

Author

Jon Awbrey, Sep 08 2005, corrected Oct 11 2005

Keywords

Comments

For positive integer m, the rote weight in gammas is g(m) = A062537(m), the rote height in gammas is h(m) = A109301(m) and the rote wayage or root degree is w(m) = omega(m) = A001221(m).

Examples

			Table of Primal Functions, Codes, Sort Parameters and Subtotals
================================================================
Primal Function | ` ` ` Primal Code ` = ` a | g h w | r | s | t
================================================================
{ } ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 1 | 0 0 0 | 1 | 1 | 1
================================================================
1:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 2 | 1 1 1 | 1 | 1 | 1
================================================================
2:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 3 | 2 2 1 | ` | ` |
1:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 4 | 2 2 1 | 2 | 2 | 2
================================================================
2:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 9 | 3 2 1 | 1 | ` |
----------------+---------------------------+-------+---+---+---
1:1 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 6 | 3 2 2 | 1 | 2 |
----------------+---------------------------+-------+---+---+---
3:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 5 | 3 3 1 | ` | ` |
4:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 7 | 3 3 1 | ` | ` |
1:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 8 | 3 3 1 | ` | ` |
1:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `16 | 3 3 1 | 4 | 4 | 6
================================================================
1:2 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `12 | 4 2 2 | ` | ` |
1:1 2:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `18 | 4 2 2 | 2 | 2 |
----------------+---------------------------+-------+---+---+---
6:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `13 | 4 3 1 | ` | ` |
9:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `23 | 4 3 1 | ` | ` |
3:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `25 | 4 3 1 | ` | ` |
2:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `27 | 4 3 1 | ` | ` |
4:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `49 | 4 3 1 | ` | ` |
1:6 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `64 | 4 3 1 | ` | ` |
2:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `81 | 4 3 1 | ` | ` |
1:9 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 512 | 4 3 1 | 8 | ` |
----------------+---------------------------+-------+---+---+---
1:1 3:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `10 | 4 3 2 | ` | ` |
1:1 4:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `14 | 4 3 2 | 2 |10 |
----------------+---------------------------+-------+---+---+---
5:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `11 | 4 4 1 | ` | ` |
7:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `17 | 4 4 1 | ` | ` |
8:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `19 | 4 4 1 | ` | ` |
1:5 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `32 | 4 4 1 | ` | ` |
16:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `53 | 4 4 1 | ` | ` |
1:7 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 128 | 4 4 1 | ` | ` |
1:8 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 256 | 4 4 1 | ` | ` |
1:16` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 65536 | 4 4 1 | 8 | 8 |20
================================================================
1:2 2:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `36 | 5 2 2 | 1 | 1 |
----------------+---------------------------+-------+---+---+---
12:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `37 | 5 3 1 | ` | ` |
18:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `61 | 5 3 1 | ` | ` |
3:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 125 | 5 3 1 | ` | ` |
6:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 169 | 5 3 1 | ` | ` |
4:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 343 | 5 3 1 | ` | ` |
9:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 529 | 5 3 1 | ` | ` |
3:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 625 | 5 3 1 | ` | ` |
2:6 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 729 | 5 3 1 | ` | ` |
4:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `2401 | 5 3 1 | ` | ` |
1:12` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `4096 | 5 3 1 | ` | ` |
2:9 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 19683 | 5 3 1 | ` | ` |
1:18` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `262144 | 5 3 1 |12 | ` |
----------------+---------------------------+-------+---+---+---
2:1 3:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `15 | 5 3 2 | ` | ` |
1:2 3:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `20 | 5 3 2 | ` | ` |
2:1 4:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `21 | 5 3 2 | ` | ` |
1:3 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `24 | 5 3 2 | ` | ` |
1:1 6:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `26 | 5 3 2 | ` | ` |
1:2 4:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `28 | 5 3 2 | ` | ` |
1:1 9:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `46 | 5 3 2 | ` | ` |
1:4 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `48 | 5 3 2 | ` | ` |
1:1 3:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `50 | 5 3 2 | ` | ` |
1:1 2:3 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `54 | 5 3 2 | ` | ` |
1:1 4:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `98 | 5 3 2 | ` | ` |
1:1 2:4 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 162 | 5 3 2 |12 |24 |
----------------+---------------------------+-------+---+---+---
10:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `29 | 5 4 1 | ` | ` |
13:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `41 | 5 4 1 | ` | ` |
14:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `43 | 5 4 1 | ` | ` |
23:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `83 | 5 4 1 | ` | ` |
25:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `97 | 5 4 1 | ` | ` |
27:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 103 | 5 4 1 | ` | ` |
5:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 121 | 5 4 1 | ` | ` |
49:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 227 | 5 4 1 | ` | ` |
2:5 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 243 | 5 4 1 | ` | ` |
7:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 289 | 5 4 1 | ` | ` |
64:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 311 | 5 4 1 | ` | ` |
8:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 361 | 5 4 1 | ` | ` |
81:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 419 | 5 4 1 | ` | ` |
1:10` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `1024 | 5 4 1 | ` | ` |
2:7 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `2187 | 5 4 1 | ` | ` |
16:2` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `2809 | 5 4 1 | ` | ` |
512:1 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `3671 | 5 4 1 | ` | ` |
2:8 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `6561 | 5 4 1 | ` | ` |
1:13` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `8192 | 5 4 1 | ` | ` |
1:14` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 16384 | 5 4 1 | ` | ` |
1:23` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` 8388608 | 5 4 1 | ` | ` |
1:25` ` ` ` ` ` | ` ` ` ` ` ` ` ` `33554432 | 5 4 1 | ` | ` |
2:16` ` ` ` ` ` | ` ` ` ` ` ` ` ` `43046721 | 5 4 1 | ` | ` |
1:27` ` ` ` ` ` | ` ` ` ` ` ` ` ` 134217728 | 5 4 1 | ` | ` |
1:49` ` ` ` ` ` | ` ` ` ` ` 562949953421312 | 5 4 1 | ` | ` |
1:64` ` ` ` ` ` | ` ` `18446744073709551616 | 5 4 1 | ` | ` |
1:81` ` ` ` ` ` | 2417851639229258349412352 | 5 4 1 | ` | ` |
1:512 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 2^512 | 5 4 1 |28 | ` |
----------------+---------------------------+-------+---+---+---
1:1 5:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `22 | 5 4 2 | ` | ` |
1:1 7:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `34 | 5 4 2 | ` | ` |
1:1 8:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `38 | 5 4 2 | ` | ` |
1:1 16:1` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 106 | 5 4 2 | 4 |32 |
----------------+---------------------------+-------+---+---+---
11:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `31 | 5 5 1 | ` | ` |
17:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `59 | 5 5 1 | ` | ` |
19:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `67 | 5 5 1 | ` | ` |
32:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 131 | 5 5 1 | ` | ` |
53:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 241 | 5 5 1 | ` | ` |
128:1 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 719 | 5 5 1 | ` | ` |
256:1 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `1619 | 5 5 1 | ` | ` |
1:11` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `2048 | 5 5 1 | ` | ` |
1:17` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `131072 | 5 5 1 | ` | ` |
1:19` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `524288 | 5 5 1 | ` | ` |
65536:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` `821641 | 5 5 1 | ` | ` |
1:32` ` ` ` ` ` | ` ` ` ` ` ` ` `4294967296 | 5 5 1 | ` | ` |
1:53` ` ` ` ` ` | ` ` ` ` `9007199254740992 | 5 5 1 | ` | ` |
1:128 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 2^128 | 5 5 1 | ` | ` |
1:256 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 2^256 | 5 5 1 | ` | ` |
1:65536 ` ` ` ` | ` ` ` ` ` ` ` ` ` 2^65536 | 5 5 1 |16 |16 |73
================================================================
a = this sequence
g = rote weight in gammas = A062537
h = rote height in gammas = A109301
w = rote wayage in gammas = A001221
r = number in (g,h,w) set = A112096
s = count in (g, h) class = A111793
t = count in weight class = A061396

Links

Crossrefs

Cf. A000079, A001221, A007097, A014221, A050924.
Cf. A061396, A062504, A062537, A062860, A106177.
Cf. A109300, A109301, A111791 to A111800, A112096.

A112096 Tetrahedron T(g, h, w) = number of rotes of weight g, height h, wayage w.

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 2, 8, 2, 8, 1, 12, 12, 28, 4, 16
Offset: 1

Views

Author

Jon Awbrey, Sep 08 2005, revised Sep 27 2005

Keywords

Comments

T(g, h, w) = |{m : A062537(m) = g, A109301(m) = h, A001221(m) = w}|.
This is the column that is labeled "r" in the tabulation of A112095.
g = h > 0 implies w = 1 and T(j, j, 1) = 2^(j-1) = A000079(j-1).

Examples

			Table T(g, h, w), omitting empty cells, starts out as follows:
g\(h,w) | (0,0) (1,1) (2,1) (2,2) (3,1) (3,2) (4,1) (4,2) (5,1)
--------+-------------------------------------------------------
0 ` ` ` | ` 1
1 ` ` ` | ` ` ` ` 1
2 ` ` ` | ` ` ` ` ` ` ` 2
3 ` ` ` | ` ` ` ` ` ` ` 1 ` ` 1 ` ` 4
4 ` ` ` | ` ` ` ` ` ` ` ` ` ` 2 ` ` 8 ` ` 2 ` ` 8
5 ` ` ` | ` ` ` ` ` ` ` ` ` ` 1 ` `12 ` `12 ` `28 ` ` 4 ` `16

Links

Crossrefs

Cf. A000079, A001221, A007097, A014221, A050924.
Cf. A061396, A062504, A062537, A062860, A106177.
Cf. A109300, A109301, A111791 to A111800, A112095.

A112869 Triangle T(g, q) = number of rotes of weight g and primal code characteristic q.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 16, 13, 56, 4
Offset: 1

Views

Author

Jon Awbrey, Oct 13 2005

Keywords

Comments

T(g, q) = |{positive integers m : A062537(m) = g and A108352(m) = q}|.
This is the column that is labeled "s" in the tabulation of A112868.
Row sums = A061396.

Examples

			Table T(g, q), omitting empty cells, begins as follows:
g\q| 0 ` 1 ` 2 ` 3 ` 4 ` 5
---+-----------------------
`0 | ` ` 1 ` ` ` ` ` ` ` `
`1 | 1 ` ` ` ` ` ` ` ` ` `
`2 | ` ` ` ` 2 ` ` ` ` ` `
`3 | 2 ` ` ` 4 ` ` ` ` ` `
`4 | 4 ` ` `16 ` ` ` ` ` `
`5 |13 ` ` `56 ` 4 ` ` ` `

Links

Crossrefs

Cf. A061396, A062504, A062537, A062860, A106177, A106178.
Cf. A108352, A108353, A108370 to A108374, A111800, A111801.
Cf. A112846, A112868.
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