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Numbers with as many even as odd prime indices, counted with multiplicity.
64

%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