A319240 Positions of zeros in A316441, the list of coefficients in the expansion of Product_{n > 1} 1/(1 + 1/n^s).

4, 6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 25, 26, 28, 33, 34, 35, 38, 39, 44, 45, 46, 48, 49, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 106, 108, 111, 112, 115, 116, 117, 118, 119, 121, 122

Offset

1,1

Comments

From Tian Vlasic, Dec 31 2021: (Start)

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

There are infinitely many terms since p^2 is a term for prime p.

Out of all numbers of the form p^k with p prime (listed in A000961), only the numbers of the form p^2 (A001248) are terms.

Out of all numbers of the form p*q^k, p and q prime, only the numbers of the form p*q (A006881), p*q^2 (A054753), p*q^4 (A178739) and p*q^6 (A189987) are terms.

Similar methods can be applied to all prime signatures. (End)

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

Example

12 = 2*6 = 3*4 = 2*2*3 has an equal number of even-length factorizations and odd-length factorizations (2). - Tian Vlasic, Dec 09 2021

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, facs[n]}], {n, 100}], 0]

Crossrefs

Complement of A319239.

Cf. A001055, A025487, A045778, A114592, A162247, A190938, A281116, A281118, A303386, A316441, A319238.

Sequence in context: A304634 A377880 A375962 * A331051 A325270 A051278

Adjacent sequences: A319237 A319238 A319239 * A319241 A319242 A319243

Keyword

nonn,easy

Author

Gus Wiseman, Sep 15 2018

Status

approved