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%I #27 Apr 17 2022 01:55:05
%S 1,6,14,15,26,33,35,36,38,51,58,65,69,74,77,84,86,90,93,95,106,119,
%T 122,123,141,142,143,145,156,158,161,177,178,185,196,198,201,202,209,
%U 210,214,215,216,217,219,221,225,226,228,249,262,265,278,287,291,299
%N Numbers with as many even as odd prime indices, counted with multiplicity.
%C These are Heinz numbers of the integer partitions counted by A045931.
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C The integers in the multiplicative subgroup of positive rational numbers generated by the products of two consecutive primes (A006094). The sequence is closed under multiplication, prime shift (A003961), and - where the result is an integer - under division. Using these closures, all the terms can be derived from the presence of 6. For example, A003961(6) = 15, A003961(15) = 35, 6 * 35 = 210, 210/15 = 14. Closed also under A297845, since A297845 can be defined using squaring, prime shift and multiplication. - _Peter Munn_, Oct 05 2020
%H David A. Corneth, <a href="/A325698/b325698.txt">Table of n, a(n) for n = 1..10000</a>
%e The sequence of terms together with their prime indices begins:
%e 1: {}
%e 6: {1,2}
%e 14: {1,4}
%e 15: {2,3}
%e 26: {1,6}
%e 33: {2,5}
%e 35: {3,4}
%e 36: {1,1,2,2}
%e 38: {1,8}
%e 51: {2,7}
%e 58: {1,10}
%e 65: {3,6}
%e 69: {2,9}
%e 74: {1,12}
%e 77: {4,5}
%e 84: {1,1,2,4}
%e 86: {1,14}
%e 90: {1,2,2,3}
%e 93: {2,11}
%e 95: {3,8}
%t Select[Range[100],Total[Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k*(-1)^PrimePi[p]]]==0&]
%o (PARI) is(n) = {my(v = vector(2), f = factor(n));for(i = 1, #f~,v[1 + primepi(f[i, 1])%2]+=f[i, 2]);v[1] == v[2]} \\ _David A. Corneth_, Oct 06 2020
%o (Python)
%o from sympy import factorint, primepi
%o def ok(n):
%o v = [0, 0]
%o for p, e in factorint(n).items(): v[primepi(p)%2] += e
%o return v[0] == v[1]
%o print([k for k in range(300) if ok(k)]) # _Michael S. Branicky_, Apr 16 2022 after _David A. Corneth_
%Y Positions of 0's in A195017.
%Y A257992(n) = A257991(n).
%Y Cf. A000712, A001222, A001405, A006094, A026010, A045931, A063886, A097613, A112798, A130780, A171966, A239241, A241638, A325700.
%Y Closed under: A003961, A003991, A297845.
%Y Subsequence of A028260, A332820.
%K nonn
%O 1,2
%A _Gus Wiseman_, May 17 2019
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