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A325698 Numbers with as many even as odd prime indices, counted with multiplicity.

%I A325698 #27 Apr 17 2022 01:55:05
%S A325698 1,6,14,15,26,33,35,36,38,51,58,65,69,74,77,84,86,90,93,95,106,119,
%T A325698 122,123,141,142,143,145,156,158,161,177,178,185,196,198,201,202,209,
%U A325698 210,214,215,216,217,219,221,225,226,228,249,262,265,278,287,291,299
%N A325698 Numbers with as many even as odd prime indices, counted with multiplicity.
%C A325698 These are Heinz numbers of the integer partitions counted by A045931.
%C A325698 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 A325698 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 A325698 David A. Corneth, <a href="/A325698/b325698.txt">Table of n, a(n) for n = 1..10000</a>
%e A325698 The sequence of terms together with their prime indices begins:
%e A325698     1: {}
%e A325698     6: {1,2}
%e A325698    14: {1,4}
%e A325698    15: {2,3}
%e A325698    26: {1,6}
%e A325698    33: {2,5}
%e A325698    35: {3,4}
%e A325698    36: {1,1,2,2}
%e A325698    38: {1,8}
%e A325698    51: {2,7}
%e A325698    58: {1,10}
%e A325698    65: {3,6}
%e A325698    69: {2,9}
%e A325698    74: {1,12}
%e A325698    77: {4,5}
%e A325698    84: {1,1,2,4}
%e A325698    86: {1,14}
%e A325698    90: {1,2,2,3}
%e A325698    93: {2,11}
%e A325698    95: {3,8}
%t A325698 Select[Range[100],Total[Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k*(-1)^PrimePi[p]]]==0&]
%o A325698 (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 A325698 (Python)
%o A325698 from sympy import factorint, primepi
%o A325698 def ok(n):
%o A325698     v = [0, 0]
%o A325698     for p, e in factorint(n).items(): v[primepi(p)%2] += e
%o A325698     return v[0] == v[1]
%o A325698 print([k for k in range(300) if ok(k)]) # _Michael S. Branicky_, Apr 16 2022 after _David A. Corneth_
%Y A325698 Positions of 0's in A195017.
%Y A325698 A257992(n) = A257991(n).
%Y A325698 Cf. A000712, A001222, A001405, A006094, A026010, A045931, A063886, A097613, A112798, A130780, A171966, A239241, A241638, A325700.
%Y A325698 Closed under: A003961, A003991, A297845.
%Y A325698 Subsequence of A028260, A332820.
%K A325698 nonn
%O A325698 1,2
%A A325698 _Gus Wiseman_, May 17 2019