A062504 -id:A062504 - cp's OEIS frontend

A061396 Number of "rooted index-functional forests" (Riffs) on n nodes. Number of "rooted odd trees with only exponent symmetries" (Rotes) on 2n+1 nodes.

1, 1, 2, 6, 20, 73, 281, 1124, 4618, 19387, 82765, 358245, 1568458, 6933765, 30907194, 138760603, 626898401, 2847946941, 13001772692, 59618918444, 274463781371, 1268064807409, 5877758070220, 27325789128330, 127384553264327, 595318139942874, 2788598203340643, 13090395266913748, 61571972632103632

Offset: 0

Jon Awbrey, Jun 09 2001

			These structures come from recursive primes' factorizations of natural numbers, where the recursion proceeds on both the exponents (^k) and the indices (_k) of the primes invoked in the factorization:
2 = (prime_1)^1 = (p_1)^1, briefly, p, weight of 1 node => a(1) = 1.
3 = (prime_2)^1 = (p_2)^1, briefly, p_p, weight of 2 nodes and
4 = (prime_1)^2 = (p_1)^2, briefly, p^p, weight of 2 nodes => a(2) = 2.

J. Awbrey, personal journal, circa 1978. Letter to N. J. A. Sloane, 1980-Aug-04.
G. Balzarotti and P. P. Lava, 103 Curiosità Matematiche, Ulrico Hoepli, Milano, Italy, 2010, pp. 269-271.

V. Jovovic, Table of n, a(n) for n=0..100
J. Awbrey, Illustration of initial terms
Jon Awbrey, Letter to N. J. A. Sloane, June 1979
Jon Awbrey, Letter to N. J. A. Sloane, August 1980
J. Awbrey, Riffs and Rotes
V. Jovovic, First 100 terms

Cf. A062504, A062860.

a(0) := 1: for k from 1 to 30 do A := add(a(i)*x^i,i=0..k): B := mul((1+x^(j+1)*A)^a(j),j=0..k-1): a(k) := coeff(series(B,x,k+1),x,k): printf(`%d,`,a(k)); od:

m = 30; a[0] = 1;
Do[A[x_] = Product[(1+x^(j+1)*Sum[a[i]*x^i, {i, 0, k}])^a[j], {j, 0, k-1}]; a[k] = SeriesCoefficient[A[x], {x, 0, k}], {k, 1, m}];
a /@ Range[0, m] (* Jean-François Alcover, Oct 19 2019 *)

G.f. A(x) = 1 + x + 2*x^2 + 6*x^3 + ... satisfies A(x) = Product_{j >= 0} (1 + x^(j+1)*A(x))^a_j.

Corrected and extended with Maple program by Vladeta Jovovic and David W. Wilson, Jun 20 2001

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

Jon Awbrey, Jun 25 2001

nonn

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

J. Awbrey, Riffs and Rotes

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

More terms from David W. Wilson, Jun 26 2001

A106177 Functional composition table for "n o m" = "n composed with m", where n and m are the "primal codes" of finite partial functions on the positive integers and 1 is the code for the empty function.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 4, 1, 1, 5, 2, 9, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 7, 1, 25, 1, 3, 1, 1, 1, 1, 1, 36, 1, 2, 1, 8, 1, 1, 1, 1, 49, 1, 5, 1, 27, 1, 1, 1, 10, 3, 1, 1, 6, 1, 1, 1, 2, 1, 1, 11, 1, 1, 2, 7, 1, 125, 4, 3, 1, 1, 1, 3, 1, 100, 1, 1, 1, 216, 1, 1, 1, 4, 1, 1, 13

Offset: 1

Jon Awbrey, May 23 2005

The right diagonal labeled by the prime power of the form j:k = (prime(j))^k contains the j^th power primes in the factorization raised to the k^th power. For example, the right diagonal labeled by the number 2 = 1:1 = (prime(1))^1 contains the power-free parts of each positive integer, specifically A055231 and the right diagonal labeled by the number 4 = 1:2 = (prime(1))^2 contains the squares of the squarefree parts of positive integers.
In general, then the right diagonal labeled by m = (j_i : k_i)_i = Product_i prime(j_i)^(k_i) contains the product over i of the (j_i)th power primes in the factorization raised to the (k_i)th powers.
For example, the operator 5 = 3:1 extracts the 3rd power primes in the factorization of each n and raises them to the first power, thus sending 8 = 1:3 to 2 = 1:1, 27 = 2:3 to 3 = 2:1 and so on.

			` ` ` ` ` ` ` ` ` ` `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
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
From _Antti Karttunen_, Nov 16 2019: (Start)
When the sequence is viewed as a square array read by falling antidiagonals, the top left 15 X 15 corner looks like this:
k=  | 1  2   3  4    5    6    7  8  9    10    11  12    13    14    15
----+--------------------------------------------------------------------
n= 1| 1, 1,  1, 1,   1,   1,   1, 1, 1,    1,    1,  1,    1,    1,    1,
   2| 1, 2,  3, 1,   5,   6,   7, 1, 1,   10,   11,  3,   13,   14,   15,
   3| 1, 1,  1, 2,   1,   1,   1, 1, 3,    1,    1,  2,    1,    1,    1,
   4| 1, 4,  9, 1,  25,  36,  49, 1, 1,  100,  121,  9,  169,  196,  225,
   5| 1, 1,  1, 1,   1,   1,   1, 2, 1,    1,    1,  1,    1,    1,    1,
   6| 1, 2,  3, 2,   5,   6,   7, 1, 3,   10,   11,  6,   13,   14,   15,
   7| 1, 1,  1, 1,   1,   1,   1, 1, 1,    1,    1,  1,    1,    1,    1,
   8| 1, 8, 27, 1, 125, 216, 343, 1, 1, 1000, 1331, 27, 2197, 2744, 3375,
   9| 1, 1,  1, 4,   1,   1,   1, 1, 9,    1,    1,  4,    1,    1,    1,
  10| 1, 2,  3, 1,   5,   6,   7, 2, 1,   10,   11,  3,   13,   14,   15,
  11| 1, 1,  1, 1,   1,   1,   1, 1, 1,    1,    1,  1,    1,    1,    1,
  12| 1, 4,  9, 2,  25,  36,  49, 1, 3,  100,  121, 18,  169,  196,  225,
  13| 1, 1,  1, 1,   1,   1,   1, 1, 1,    1,    1,  1,    1,    1,    1,
  14| 1, 2,  3, 1,   5,   6,   7, 1, 1,   10,   11,  3,   13,   14,   15,
  15| 1, 1,  1, 2,   1,   1,   1, 2, 3,    1,    1,  2,    1,    1,    1,
(End)

Antti Karttunen, Table of n, a(n) for n = 1..25425; the first 225 antidiagonals of the array
J. Awbrey, Riffs and Rotes

Cf. A000040, A061396, A062504, A062537, A062860, A106178, A181819, A286561.
Rows 1 - 3: A000012, A055231, A329376.
Columns 1 - 2: A000012, A006519.

up_to = 105;
A106177sq(n,k) = { my(f = factor(k)); prod(i=1,#f~,f[i, 1]^valuation(n, prime(f[i, 2]))); };
A106177list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A106177sq(col,(a-(col-1))))); (v); };
v106177 = A106177list(up_to);
A106177(n) = v106177[n]; \\ Antti Karttunen, Nov 16 2019

If k = Product p_i^e_i, A(n,k) = p_i^A286561(n, A000040(e_i)), where A286561(x,y) gives the y-valuation of x. - Antti Karttunen, Nov 16 2019

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

Jon Awbrey, Jun 24 2005 - Jul 08 2005

nonn

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.

			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.

J. Awbrey, Riffs and Rotes

Cf. A007097, A050924, A061396, A062504, A062537, A062860.

Cf. A106177, A108352, A108371, A109300, A111791 to A111800.

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

Jon Awbrey, Jul 04 2005, revised Sep 06 2005

nonn

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.

			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` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

J. Awbrey, Riffs and Rotes

Cf. A007097, A050924, A061396, A062504, A062537, A062860.

Cf. A109301, A106177, A108352, A108371, A111791 to A111800.

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

Jon Awbrey, Aug 24 2005, revised Sep 02 2005

			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).

J. Awbrey, Riffs and Rotes

Cf. A007097, A050924, A061396, A062504, A062537, A062860.

Cf. A109300, A109301, A111792 to A111800.

A108352 a(n) = primal code characteristic of n, which is the least positive integer, if any, such that (n o)^k = 1, otherwise equal to 0. Here "o" denotes the primal composition operator, as illustrated in A106177 and A108371 and (n o)^k = n o ... o n, with k occurrences of n.

Original entry on oeis.org

1, 0, 2, 2, 2, 0, 2, 2, 0, 0, 2, 0, 2, 0, 2, 2, 2, 0, 2, 3, 2, 0, 2, 3, 2, 0, 2, 3, 2, 0, 2, 2, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 3, 0, 0, 2, 3, 2, 0, 2, 3, 2, 0, 2, 3, 2, 0, 2, 0, 2, 0, 0, 2, 2, 0, 2, 3, 2, 0, 2, 0, 2, 0, 3, 3, 2, 0, 2, 3, 2, 0, 2, 0, 2, 0, 2, 3, 2, 0, 2, 3, 2, 0, 2, 3, 2, 0, 0, 2, 2, 0, 2, 3, 2

Offset: 1

Jon Awbrey, May 31 2005, revised Jun 01 2005

nonn

			a(1) = 1 because (1 o)^1 = ({ } o)^1 = 1.
a(2) = 0 because (2 o)^k = (1:1 o)^k = 2, for all positive k.
a(3) = 2 because (3 o)^2 = (2:1 o)^2 = 1.
a(4) = 2 because (4 o)^2 = (1:2 o)^2 = 1.
a(5) = 2 because (5 o)^2 = (3:1 o)^2 = 1.
a(6) = 0 because (6 o)^k = (1:1 2:1 o)^k = 6, for all positive k.
a(7) = 2 because (7 o)^2 = (4:1 o)^1 = 1.
a(8) = 2 because (8 o)^2 = (1:3 o)^1 = 1.
a(9) = 0 because (9 o)^k = (2:2 o)^k = 9, for all positive k.
a(10) = 0 because (10 o)^k = (1:1 3:1 o)^k = 10, for all positive k.
Detail of calculation for compositional powers of 12:
(12 o)^2 = (1:2 2:1) o (1:2 2:1) = (1:1 2:2) = 18
(12 o)^3 = (1:1 2:2) o (1:2 2:1) = (1:2 2:1) = 12
Detail of calculation for compositional powers of 20:
(20 o)^2 = (1:2 3:1) o (1:2 3:1) = (3:2) = 25
(20 o)^3 = (3:2) o (1:2 3:1) = 1.
From _Antti Karttunen_, Nov 20 2019: (Start)
For n=718, because 718 = prime(1)^1 * prime(72)^1, its partial function primal code is (1:1 72:1), which, when composed with itself stays same (that is, A106177(718,718) = 718), thus, as 1 is never reached, a(718) = 0, like is true for all even nonsquare semiprimes.
For n=1804, as 1804 = prime(1)^2 * prime(5)^1 * prime(13)^1, its primal code is (1:2 5:1 13:1), which, when composed with itself yields 203401 = prime(5)^2 * prime(13)^2, i.e., primal code (5:2 13:2), which when composed with (1:2 5:1 13:1) yields 1, which happened on the second iteration, thus a(1804) = 2+1 = 3.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..100000
Jon Awbrey, Riffs and Rotes
Jon Awbrey, Primal Code Characteristic, n = 1 to 3000 (Note: values given here differ at n = 718, 746, 1156, 1449, 1734 and 1804 from those computed in b-file). - _Antti Karttunen_, Nov 23 2019

Cf. A001747, A046315, A055231, A061396, A062504, A062537, A062860, A065091, A106177, A106178.

Cf. A108353, A108370, A108371, A108372, A108373, A108374, A111801.

A106177sq(n,k) = { my(f = factor(k)); prod(i=1,#f~,f[i, 1]^valuation(n, prime(f[i, 2]))); }; \\ As in A106177.
A108352(n) = { my(orgn=n,xs=Set([]), k=1); while(n>1, if(vecsearch(xs,n), return(0)); xs = setunion([n],xs); n = A106177sq(n,orgn); k++); (k); }; \\ Antti Karttunen, Nov 20 2019

a(A065091(n)) = 2 for all n, a(A001747(n)) = 0 for all n, except n=2, and a(A046315(n)) = 2 for n > 1. - Antti Karttunen, Nov 20 2019

Links and cross-references added, Aug 19 2005

Term a(63) corrected and five more terms added (up to a(105)) by Antti Karttunen, Nov 20 2019

A111800 Order of the rote (rooted odd tree with only exponent symmetries) for n.

Original entry on oeis.org

1, 3, 5, 5, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 11, 7, 9, 9, 9, 11, 11, 11, 9, 11, 9, 11, 9, 11, 11, 13, 11, 9, 13, 11, 13, 11, 11, 11, 13, 13, 11, 13, 11, 13, 13, 11, 13, 11, 9, 11, 13, 13, 9, 11, 15, 13, 13, 13, 11, 15, 11, 13, 13, 9, 15, 15, 11, 13, 13, 15, 13, 13, 13, 13, 13, 13, 15, 15

Offset: 1

Jon Awbrey, Aug 17 2005, based on calculations by David W. Wilson

A061396(n) gives the number of times that 2n+1 appears in this sequence.

			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.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
J. Awbrey, Illustrations of Rotes for Small Integers
J. Awbrey, Riffs and Rotes

Cf. A061396, A062504, A062537, A062860, A106177, A109300, A109301.

with(numtheory):
a:= proc(n) option remember;
      1+add(a(pi(i[1]))+a(i[2]), i=ifactors(n)[2])
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 25 2015

a[1] = 1; a[n_] := a[n] = 1+Sum[a[PrimePi[i[[1]] ] ] + a[i[[2]] ], {i, FactorInteger[n]}]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

a(Prod(p_i^e_i)) = 1 + Sum(a(i) + a(e_i)), product over nonzero e_i in prime factorization of n.

A106178 Functional composition table for "n o m" = "n composed with m", where n and m are the "primal codes" of finite partial functions on the positive integers and 1 is the code for the empty function, but omitting the trivial values of 1 at the margins of the table.

Original entry on oeis.org

2, 3, 1, 1, 1, 4, 5, 2, 9, 1, 6, 1, 1, 1, 2, 7, 1, 25, 1, 3, 1, 1, 1, 36, 1, 2, 1, 8, 1, 1, 49, 1, 5, 1, 27, 1, 10, 3, 1, 1, 6, 1, 1, 1, 2, 11, 1, 1, 2, 7, 1, 125, 4, 3, 1, 3, 1, 100, 1, 1, 1, 216, 1, 1, 1, 4, 13, 2, 121, 1, 3, 1, 343, 1, 5, 1, 9, 1, 14, 1, 9, 1, 10, 1, 1, 1, 6, 1, 2, 1, 2

Offset: 2

Jon Awbrey, May 28 2005

This sequence is derived from A106177 by removing the "obvious" values of 1 at the margins of the triangular array.

			` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` n o m
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 1 . 1
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 2 . ` . 2
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 3 . ` . ` . 3
` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` 4 . ` . 2 . ` . 4
` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` 5 . ` . 3 . 1 . ` . 5
` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` 6 . ` . 1 . 1 . 4 . ` . 6
` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` 7 . ` . 5 . 2 . 9 . 1 . ` . 7
` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` 8 . ` . 6 . 1 . 1 . 1 . 2 . ` . 8
` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` 9 . ` . 7 . 1 . 25. 1 . 3 . 1 . ` . 9
` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` `10 . ` . 1 . 1 . 36. 1 . 2 . 1 . 8 . ` . 10
` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` `11 . ` . 1 . 1 . 49. 1 . 5 . 1 . 27. 1 . ` . 11
` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` `12 . ` . 10. 3 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . ` . 12
` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` `13 . ` . 11. 1 . 1 . 2 . 7 . 1 .125. 4 . 3 . 1 . ` . 13
` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` `14 . ` . 3 . 1 .100. 1 . 1 . 1 .216. 1 . 1 . 1 . 4 . ` . 14
` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` `15 . ` . 13. 2 .121. 1 . 3 . 1 .343. 1 . 5 . 1 . 9 . 1 . ` . 15
` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` `16 . ` . 14. 1 . 9 . 1 . 10. 1 . 1 . 1 . 6 . 1 . 2 . 1 . 2 . ` . 16

J. Awbrey, Riffs and Rotes

Cf. A106177, A061396, A062504, A062537, A062860.

cp's OEIS Frontend

A061396 Number of "rooted index-functional forests" (Riffs) on n nodes. Number of "rooted odd trees with only exponent symmetries" (Rotes) on 2n+1 nodes.

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

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A106177 Functional composition table for "n o m" = "n composed with m", where n and m are the "primal codes" of finite partial functions on the positive integers and 1 is the code for the empty function.

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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.

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

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

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A108352 a(n) = primal code characteristic of n, which is the least positive integer, if any, such that (n o)^k = 1, otherwise equal to 0. Here "o" denotes the primal composition operator, as illustrated in A106177 and A108371 and (n o)^k = n o ... o n, with k occurrences of n.

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A111800 Order of the rote (rooted odd tree with only exponent symmetries) for n.

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