A108352 -id:A108352 - cp's OEIS frontend

A108370 Numbers whose primal code characteristic = 0, that is, positive n for which A108352(n) = 0.

2, 6, 9, 10, 12, 14, 18, 22, 26, 30, 34, 36, 38, 40, 42, 45, 46, 50, 54, 58, 60, 62, 66, 70, 72, 74, 78, 82, 84, 86, 90, 94, 98, 99, 102, 106, 110, 112, 114, 117, 118, 120, 122, 125, 126, 130, 132, 134, 138, 142, 144, 146, 150, 153, 154, 156, 158, 162, 166, 170, 171

Offset: 1

Jon Awbrey, Jun 01 2005

nonn

J. Awbrey, Riffs and Rotes

Cf. A108352, A108371, A108372, A108373, A108374.

A108374 Numbers whose primal code characteristic = 4, that is, positive n for which A108352(n) = 4.

Original entry on oeis.org

756, 1176, 1188, 1200, 1400, 1404, 1620, 1836, 2052, 2160, 2200, 2400, 2484, 2600, 2904

Offset: 1

Jon Awbrey, Jun 16 2005, extended Jul 10 2005

nonn

			Writing (prime(i))^j as i:j, we have the following table:
Primal Functions and Functional Digraphs for a(1) to a(15)
0756 = 1:2 2:3 4:1 || 4 -> 1 -> 2 -> 3
1176 = 1:3 2:1 4:2 || 4 -> 2 -> 1 -> 3
1188 = 1:2 2:3 5:1 || 5 -> 1 -> 2 -> 3
1200 = 1:4 2:1 3:2 || 3 -> 2 -> 1 -> 4
1400 = 1:3 3:2 4:1 || 4 -> 1 -> 3 -> 2
1404 = 1:2 2:3 6:1 || 6 -> 1 -> 2 -> 3
1620 = 1:2 2:4 3:1 || 3 -> 1 -> 2 -> 4
1836 = 1:2 2:3 7:1 || 7 -> 1 -> 2 -> 3
2052 = 1:2 2:3 8:1 || 8 -> 1 -> 2 -> 3
2160 = 1:4 2:3 3:1 || 2 -> 3 -> 1 -> 4
2200 = 1:3 3:2 5:1 || 5 -> 1 -> 3 -> 2
2400 = 1:5 2:1 3:2 || 3 -> 2 -> 1 -> 5
2484 = 1:2 2:3 9:1 || 9 -> 1 -> 2 -> 3
2600 = 1:3 3:2 6:1 || 6 -> 1 -> 3 -> 2
2904 = 1:3 2:1 5:2 || 5 -> 2 -> 1 -> 3

J. Awbrey, Riffs and Rotes

Cf. A108352, A108353, A108370, A108371, A108372, A108373.

A108353 For each nonnegative integer n, a(n) is the smallest positive integer j whose primal code characteristic is n, that is, the smallest j such that A108352(j) = n.

Original entry on oeis.org

2, 1, 3, 20, 756, 178200

Offset: 0

Jon Awbrey, Jun 17 2005, extended Aug 20 2005

nonn

Suggested by Antti Karttunen.

			Writing (prime(i))^j as i:j, we have the following table:
Primal Functions and Functional Digraphs for a(0) to a(5)
` ` ` 2 = 1:1 ` ` ` ` ` ` || 1 -> 1 (infinite loop)
` ` ` 1 = { } ` ` ` ` ` ` || 1
` ` ` 3 = 2:1 ` ` ` ` ` ` || 2 -> 1
` ` `20 = 1:2 3:1 ` ` ` ` || 3 -> 1 -> 2
` ` 756 = 1:2 2:3 4:1 ` ` || 4 -> 1 -> 2 -> 3
`178200 = 1:3 2:4 3:2 5:1 || 5 -> 1 -> 3 -> 2 -> 4

J. Awbrey, Riffs and Rotes

Cf. A106177, A108352, A108370, A108371, A108372, A108373, A108374, A111801.

A111801 Numbers that have a positive primal code characteristic, that is, positive integers j for which A108352(j) > 0.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 11, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 37, 39, 41, 43, 44, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 61, 63, 64, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 85, 87, 88, 89, 91, 92, 93, 95, 96, 97, 100, 101, 103, 104, 105

Offset: 1

Jon Awbrey, Aug 19 2005

nonn

This set of numbers is the complement of the set in A108370.

J. Awbrey, Riffs and Rotes

Cf. A106177, A108352, A108370, A108371, A108372, A108373, A108374.

A108372 Numbers whose primal code characteristic = 2, that is, positive n for which A108352(n) = 2.

Original entry on oeis.org

3, 4, 5, 7, 8, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 100, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 127, 128, 129, 131

Offset: 1

Jon Awbrey, Jun 01 2005

nonn

J. Awbrey, Riffs and Rotes

Cf. A108352, A108370, A108371, A108373, A108374.

A108373 Numbers whose primal code characteristic = 3, that is, positive n for which A108352(n) = 3.

Original entry on oeis.org

20, 24, 28, 44, 48, 52, 56, 68, 75, 76, 80, 88, 92, 96, 104, 108, 116, 124, 135, 136, 140, 147, 148, 152, 160, 164, 168, 172, 176, 184, 188, 192, 200, 208, 212, 220, 224, 232, 236, 240, 244, 248, 260, 264, 268, 272, 284, 292, 296, 304, 308, 312, 316, 320, 324

Offset: 1

Jon Awbrey, Jun 01 2005, extended Jun 30 2005

nonn

J. Awbrey, Riffs and Rotes

Cf. A108352, A108353, A108370, A108371, A108372, A108374.

A109298 Primal codes of finite idempotent functions on positive integers.

Original entry on oeis.org

1, 2, 9, 18, 125, 250, 1125, 2250, 2401, 4802, 21609, 43218, 161051, 300125, 322102, 600250, 1449459, 2701125, 2898918, 4826809, 5402250, 9653618, 20131375, 40262750, 43441281, 86882562, 181182375, 362364750, 386683451, 410338673, 603351125, 773366902, 820677346

Offset: 1

Jon Awbrey, Jul 06 2005

nonn

Finite idempotent functions are identity maps on finite subsets, counting the empty function as the idempotent on the empty set.
From Gus Wiseman, Mar 09 2019: (Start)
Also numbers whose ordered prime signature is equal to the distinct prime indices in increasing order. A prime index of n is a number m such that prime(m) divides n. The ordered prime signature (A124010) is the sequence of multiplicities (or exponents) in a number's prime factorization, taken in order of the prime base. The case where the prime indices are taken in decreasing order is A324571.
Also numbers divisible by prime(k) exactly k times for each prime index k. These are a kind of self-describing numbers (cf. A001462, A304679).
Also Heinz numbers of integer partitions where the multiplicity of m is m for all m in the support (counted by A033461). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also products of distinct elements of A062457. For example, 43218 = prime(1)^1 * prime(2)^2 * prime(4)^4.
(End)

			Writing (prime(i))^j as i:j, we have the following table of examples:
Primal Codes of Finite Idempotent Functions on Positive Integers
` ` ` 1 = { }
` ` ` 2 = 1:1
` ` ` 9 = ` ` 2:2
` ` `18 = 1:1 2:2
` ` 125 = ` ` ` ` 3:3
` ` 250 = 1:1 ` ` 3:3
` `1125 = ` ` 2:2 3:3
` `2250 = 1:1 2:2 3:3
` `2401 = ` ` ` ` ` ` 4:4
` `4802 = 1:1 ` ` ` ` 4:4
` 21609 = ` ` 2:2 ` ` 4:4
` 43218 = 1:1 2:2 ` ` 4:4
`161051 = ` ` ` ` ` ` ` ` 5:5
`300125 = ` ` ` ` 3:3 4:4
`322102 = 1:1 ` ` ` ` ` ` 5:5
`600250 = 1:1 ` ` 3:3 4:4
From _Gus Wiseman_, Mar 09 2019: (Start)
The sequence of terms together with their prime indices begins as follows. For example, we have 18: {1,2,2} because 18 = prime(1) * prime(2) * prime(2) has prime signature {1,2} and the distinct prime indices are also {1,2}.
       1: {}
       2: {1}
       9: {2,2}
      18: {1,2,2}
     125: {3,3,3}
     250: {1,3,3,3}
    1125: {2,2,3,3,3}
    2250: {1,2,2,3,3,3}
    2401: {4,4,4,4}
    4802: {1,4,4,4,4}
   21609: {2,2,4,4,4,4}
   43218: {1,2,2,4,4,4,4}
  161051: {5,5,5,5,5}
  300125: {3,3,3,4,4,4,4}
  322102: {1,5,5,5,5,5}
  600250: {1,3,3,3,4,4,4,4}
(End)

David A. Corneth, Table of n, a(n) for n = 1..10000
J. Awbrey, Riffs and Rotes

Cf. A076954, A106177, A108352, A108371, A109297, A109301.
Cf. A001156, A033461, A056239, A062457, A112798, A118914, A124010 (ordered prime signature), A181819, A276078, A304679.
Cf. A324524, A324525, A324570, A324571, A324572, A324587, A324588.
Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

Select[Range[10000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]==k]&]

is(n) = my(f = factor(n)); for(i = 1, #f~, if(prime(f[i, 2]) != f[i, 1], return(0))); 1 \\ David A. Corneth, Mar 09 2019

Sum_{n>=1} 1/a(n) = Product_{n>=1} (1 + 1/prime(n)^n) = 1.6807104966... - Amiram Eldar, Jan 03 2021

Offset set to 1, missing terms inserted and more terms added by Alois P. Heinz, Mar 08 2019

A109297 Primal codes of finite permutations on positive integers.

Original entry on oeis.org

1, 2, 9, 12, 18, 40, 112, 125, 250, 352, 360, 540, 600, 675, 832, 1008, 1125, 1350, 1500, 2176, 2250, 2268, 2352, 2401, 3168, 3969, 4802, 4864, 7488, 7938, 10692, 11616, 11776, 14000, 19584, 21609, 27440, 28812, 29403, 29696, 32448, 35000, 37908, 43218, 43776

Offset: 1

Jon Awbrey, Jul 08 2005

nonn

A finite permutation is a bijective mapping from a finite set to itself, counting the empty mapping as a permutation of the empty set.

Also Heinz numbers of integer partitions where the set of distinct parts is equal to the set of distinct multiplicities. These partitions are counted by A114640. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 02 2019

			Writing (prime(i))^j as i:j, we have the following table:
Primal Codes of Finite Permutations on Positive Integers
` ` ` 1 = { }
` ` ` 2 = 1:1
` ` ` 9 = 2:2
` ` `12 = 1:2 2:1
` ` `18 = 1:1 2:2
` ` `40 = 1:3 3:1
` ` 112 = 1:4 4:1
` ` 125 = 3:3
` ` 250 = 1:1 3:3
` ` 352 = 1:5 5:1
` ` 360 = 1:3 2:2 3:1
` ` 540 = 1:2 2:3 3:1
` ` 600 = 1:3 2:1 3:2
` ` 675 = 2:3 3:2
` ` 832 = 1:6 6:1
` `1008 = 1:4 2:2 4:1
` `1125 = 2:2 3:3
` `1350 = 1:1 2:3 3:2
` `1500 = 1:2 2:1 3:3
` `2176 = 1:7 7:1
` `2250 = 1:1 2:2 3:3

Amiram Eldar, Table of n, a(n) for n = 1..300 (terms 1..100 from Alois P. Heinz)
Jon Awbrey, Riffs and Rotes.

Subsequence of A130091.
Cf. A076954, A106177, A108352, A108371, A109298, A109301, A056239, A112798, A114640, A118914.
Cf. A324524, A324525, A324571, A325127, A325128, A325130, A325131.

a:= proc(n) option remember; local k; for k from 1+`if`(n=1, 0,
      a(n-1)) while (l-> sort(map(i-> i[2], l)) <> sort(map(
      i-> numtheory[pi](i[1]), l)))(ifactors(k)[2]) do od; k
    end:
seq(a(n), n=1..45);  # Alois P. Heinz, Mar 08 2019

Select[Range[1000],#==1||Union[PrimePi/@First/@FactorInteger[#]]==Union[Last/@FactorInteger[#]]&] (* Gus Wiseman, Apr 02 2019 *)

is(n) = {my(f = factor(n), p = f[,1], e = vecsort(f[,2])); for(i=1, #p, if(primepi(p[i]) != e[i], return(0))); 1}; \\ Amiram Eldar, Jul 30 2022

More terms from Franklin T. Adams-Watters, Dec 19 2005

Offset set to 1 by Alois P. Heinz, Mar 08 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).

cp's OEIS Frontend

A108370 Numbers whose primal code characteristic = 0, that is, positive n for which A108352(n) = 0.

Views

Author

Keywords

Links

Crossrefs

A108374 Numbers whose primal code characteristic = 4, that is, positive n for which A108352(n) = 4.

Views

Author

Keywords

Examples

Links

Crossrefs

A108353 For each nonnegative integer n, a(n) is the smallest positive integer j whose primal code characteristic is n, that is, the smallest j such that A108352(j) = n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A111801 Numbers that have a positive primal code characteristic, that is, positive integers j for which A108352(j) > 0.

Views

Author

Keywords

Comments

Links

Crossrefs

A108372 Numbers whose primal code characteristic = 2, that is, positive n for which A108352(n) = 2.

Views

Author

Keywords

Links

Crossrefs

A108373 Numbers whose primal code characteristic = 3, that is, positive n for which A108352(n) = 3.

Views

Author

Keywords

Links

Crossrefs

A109298 Primal codes of finite idempotent functions on positive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A109297 Primal codes of finite permutations on positive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords