A109298 Primal codes of finite idempotent functions on positive integers.

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

1,2

Comments

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)

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

J. Awbrey, Riffs and Rotes

Formula

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

Example

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)

Mathematica

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

Prog

(PARI) 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

Crossrefs

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.

Sequence in context: A083423 A068978 A006226 * A297470 A075537 A342473

Adjacent sequences: A109295 A109296 A109297 * A109299 A109300 A109301

Keyword

nonn

Author

Jon Awbrey, Jul 06 2005

Extensions

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

Status

approved