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

6, 8, 10, 14, 15, 16, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 64, 65, 69, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 96, 106, 111, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 144, 145, 146, 155, 158, 159, 160, 161, 166

Offset: 1

Author

Gus Wiseman, Sep 15 2018

Keywords

• nonn

Comments

From Tian Vlasic, Jan 01 2022: (Start)

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

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

For primes p and q, p*q^k is a term if and only if k = A000326(m)+N with 0 <= N < m. (End)

Examples

			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
		

Links

• Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
• Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 2.
• Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
• Wikipedia, Pentagonal number theorem

Crossrefs

Complement of A319237.

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

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Join@@Position[Table[Sum[(-1)^Length[f],{f,Select[facs[n],UnsameQ@@#&]}],{n,100}],0]