A319240 - OEIS

%I #55 Feb 09 2022 21:23:10

%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,

%T 50,51,52,55,57,58,62,63,65,68,69,72,74,75,76,77,80,82,85,86,87,91,92,

%U 93,94,95,98,99,106,108,111,112,115,116,117,118,119,121,122

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

%C From _Tian Vlasic_, Dec 31 2021: (Start)

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

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

%C 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.

%C 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.

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

%H David A. Corneth, <a href="/A319240/b319240.txt">Table of n, a(n) for n = 1..10000</a>

%e 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

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

%Y Complement of A319239.

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

%K nonn,easy

%O 1,1

%A _Gus Wiseman_, Sep 15 2018