A353374 - OEIS

%I #14 Nov 03 2022 16:35:34

%S 1,0,0,1,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,

%T 1,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,1,0,1,0,0,1,0,0,0,1,1,0,0,0,

%U 1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,1,1,0,1,0,0,0,1,0,1,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,1

%N a(n) = 1 if the prime factorization of n has an even number of prime factors that sum to an even number, otherwise 0.

%C This is the characteristic function of A345452, see comments there.

%H Antti Karttunen, <a href="/A353374/b353374.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

%F a(n) = A035263(n) * A065043(n).

%F a(n) = [A008836(n) == +1] * [A001414(n) is even], where [ ] is the Iverson bracket.

%F a(n) = A065043(n) * A356163(n). - _Antti Karttunen_, Nov 03 2022

%o (PARI) A353374(n) = if(bigomega(n)%2,0, my(f = factor(n)); 0==((sum(i=1, #f~, f[i,2]*f[i, 1]))%2));

%o (PARI) A353374(n) = (!(bigomega(n)%2) && !(valuation(n, 2)%2)); \\ After the function "is" provided in A345452 by _David A. Corneth_, Jun 24 2021

%Y Characteristic function of A345452.

%Y Cf. A001222, A001414, A008836, A035263, A065043, A353375, A353376 (inverse Möbius transform), A353377, A353378, A356163.

%Y Cf. also A353331, A353370.

%K nonn

%O 1

%A _Antti Karttunen_, Apr 17 2022