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A319238 Positions of zeros in A114592, the list of coefficients in the expansion of Product_{n > 1} (1 - 1/n^s).

%I A319238 #53 Apr 03 2025 01:36:02
%S A319238 6,8,10,14,15,16,21,22,26,27,33,34,35,38,39,46,51,55,57,58,62,64,65,
%T A319238 69,74,77,81,82,85,86,87,91,93,94,95,96,106,111,115,118,119,120,122,
%U A319238 123,125,129,133,134,141,142,143,144,145,146,155,158,159,160,161,166
%N A319238 Positions of zeros in A114592, the list of coefficients in the expansion of Product_{n > 1} (1 - 1/n^s).
%C A319238 From _Tian Vlasic_, Jan 01 2022: (Start)
%C A319238 Numbers that have an equal number of even- and odd-length unordered factorizations into distinct factors.
%C A319238 For prime p, by the pentagonal number theorem, p^k is a term if and only if k is in A090864.
%C A319238 For primes p and q, p*q^k is a term if and only if k = A000326(m)+N with 0 <= N < m. (End)
%H A319238 Leonhard Euler, <a href="https://arxiv.org/abs/math/0505373">On the remarkable properties of the pentagonal numbers</a>, arXiv:math/0505373 [math.HO], 2005.
%H A319238 Leonhard Euler, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k6953p/f517">De mirabilibus proprietatibus numerorum pentagonalium</a>, par. 2.
%H A319238 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalNumberTheorem.html">Pentagonal Number Theorem</a>
%H A319238 Wikipedia, <a href="http://www.wikipedia.org/wiki/Pentagonal number theorem">Pentagonal number theorem</a>
%e A319238 16 = 2*8 = 4*4 = 2*2*4 = 2*2*2*2 has an equal number of even-length factorizations and odd-length factorizations into distinct factors (1). - _Tian Vlasic_, Dec 31 2021
%t A319238 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A319238 Join@@Position[Table[Sum[(-1)^Length[f],{f,Select[facs[n],UnsameQ@@#&]}],{n,100}],0]
%Y A319238 Complement of A319237.
%Y A319238 Cf. A001055, A045778, A114592, A162247, A190938, A281116, A281118, A303386, A316441, A319240.
%K A319238 nonn
%O A319238 1,1
%A A319238 _Gus Wiseman_, Sep 15 2018