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Positions of zeros in A114592, the list of coefficients in the expansion of Product_{n > 1} (1 - 1/n^s).
9

%I #47 Feb 09 2022 21:26:33

%S 6,8,10,14,15,16,21,22,26,27,33,34,35,38,39,46,51,55,57,58,62,64,65,

%T 69,74,77,81,82,85,86,87,91,93,94,95,96,106,111,115,118,119,120,122,

%U 123,125,129,133,134,141,142,143,144,145,146,155,158,159,160,161,166

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

%C From _Tian Vlasic_, Jan 01 2022: (Start)

%C Numbers that have an equal number of even- and odd-length unordered factorizations into distinct factors.

%C For prime p, by the pentagonal number theorem, p^k is a term if and only if k is in A090864.

%C 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 L. 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 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E542.html">De mirabilibus proprietatibus numerorum pentagonalium</a>, par. 2

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumberTheorem.html">Pentagonal Number Theorem</a>

%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Pentagonal number theorem">Pentagonal number theorem</a>

%e 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 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Join@@Position[Table[Sum[(-1)^Length[f],{f,Select[facs[n],UnsameQ@@#&]}],{n,100}],0]

%Y Complement of A319237.

%Y Cf. A001055, A045778, A114592, A162247, A190938, A281116, A281118, A303386, A316441, A319240.

%K nonn

%O 1,1

%A _Gus Wiseman_, Sep 15 2018