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%I #28 Jan 03 2021 02:54:17
%S 1,2,9,18,125,250,1125,2250,2401,4802,21609,43218,161051,300125,
%T 322102,600250,1449459,2701125,2898918,4826809,5402250,9653618,
%U 20131375,40262750,43441281,86882562,181182375,362364750,386683451,410338673,603351125,773366902,820677346
%N Primal codes of finite idempotent functions on positive integers.
%C Finite idempotent functions are identity maps on finite subsets, counting the empty function as the idempotent on the empty set.
%C From _Gus Wiseman_, Mar 09 2019: (Start)
%C 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.
%C 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).
%C 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).
%C Also products of distinct elements of A062457. For example, 43218 = prime(1)^1 * prime(2)^2 * prime(4)^4.
%C (End)
%H David A. Corneth, <a href="/A109298/b109298.txt">Table of n, a(n) for n = 1..10000</a>
%H J. Awbrey, <a href="https://oeis.org/wiki/Riffs_and_Rotes">Riffs and Rotes</a>
%F Sum_{n>=1} 1/a(n) = Product_{n>=1} (1 + 1/prime(n)^n) = 1.6807104966... - _Amiram Eldar_, Jan 03 2021
%e Writing (prime(i))^j as i:j, we have the following table of examples:
%e Primal Codes of Finite Idempotent Functions on Positive Integers
%e ` ` ` 1 = { }
%e ` ` ` 2 = 1:1
%e ` ` ` 9 = ` ` 2:2
%e ` ` `18 = 1:1 2:2
%e ` ` 125 = ` ` ` ` 3:3
%e ` ` 250 = 1:1 ` ` 3:3
%e ` `1125 = ` ` 2:2 3:3
%e ` `2250 = 1:1 2:2 3:3
%e ` `2401 = ` ` ` ` ` ` 4:4
%e ` `4802 = 1:1 ` ` ` ` 4:4
%e ` 21609 = ` ` 2:2 ` ` 4:4
%e ` 43218 = 1:1 2:2 ` ` 4:4
%e `161051 = ` ` ` ` ` ` ` ` 5:5
%e `300125 = ` ` ` ` 3:3 4:4
%e `322102 = 1:1 ` ` ` ` ` ` 5:5
%e `600250 = 1:1 ` ` 3:3 4:4
%e From _Gus Wiseman_, Mar 09 2019: (Start)
%e 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}.
%e 1: {}
%e 2: {1}
%e 9: {2,2}
%e 18: {1,2,2}
%e 125: {3,3,3}
%e 250: {1,3,3,3}
%e 1125: {2,2,3,3,3}
%e 2250: {1,2,2,3,3,3}
%e 2401: {4,4,4,4}
%e 4802: {1,4,4,4,4}
%e 21609: {2,2,4,4,4,4}
%e 43218: {1,2,2,4,4,4,4}
%e 161051: {5,5,5,5,5}
%e 300125: {3,3,3,4,4,4,4}
%e 322102: {1,5,5,5,5,5}
%e 600250: {1,3,3,3,4,4,4,4}
%e (End)
%t Select[Range[10000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]==k]&]
%o (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
%Y Cf. A076954, A106177, A108352, A108371, A109297, A109301.
%Y Cf. A001156, A033461, A056239, A062457, A112798, A118914, A124010 (ordered prime signature), A181819, A276078, A304679.
%Y Cf. A324524, A324525, A324570, A324571, A324572, A324587, A324588.
%Y Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.
%K nonn
%O 1,2
%A _Jon Awbrey_, Jul 06 2005
%E Offset set to 1, missing terms inserted and more terms added by _Alois P. Heinz_, Mar 08 2019
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