cp's OEIS Frontend

A360158 a(n) is the number of unitary divisors of n that are odd squares minus the number of unitary divisors of n that are even squares.

%I A360158 #9 Jan 29 2023 12:02:33
%S A360158 1,1,1,0,1,1,1,1,2,1,1,0,1,1,1,0,1,2,1,0,1,1,1,1,2,1,1,0,1,1,1,1,1,1,
%T A360158 1,0,1,1,1,1,1,1,1,0,2,1,1,0,2,2,1,0,1,1,1,1,1,1,1,0,1,1,2,0,1,1,1,0,
%U A360158 1,1,1,2,1,1,2,0,1,1,1,0,2,1,1,0,1,1,1
%N A360158 a(n) is the number of unitary divisors of n that are odd squares minus the number of unitary divisors of n that are even squares.
%C A360158 The unitary analog of A344299.
%C A360158 The least term that is larger than 2 is a(225) = 4.
%H A360158 Amiram Eldar, <a href="/A360158/b360158.txt">Table of n, a(n) for n = 1..10000</a>
%F A360158 a(n) = Sum_{d|n, gcd(d, n/d)=1, d square} (-1)^(d+1).
%F A360158 Multiplicative with a(2^e) = 1 if e is odd and 0 if e is even, and for p > 2, a(p^e) = 1 if e is odd and 2 if e is even.
%F A360158 Dirichlet g.f.: (zeta(s)*zeta(2*s)/zeta(3*s)) * (4^s + 2^s - 1)/(4^s + 2^s + 1).
%F A360158 Sum_{k=1..n} a(k) ~ c * n, where c = 5*zeta(2)/(7*zeta(3)) = 0.977451984014... .
%t A360158 f[p_, e_] := If[OddQ[e], 1, 2]; f[2, e_] := If[OddQ[e], 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A360158 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, 1, if(f[i, 1] == 2, 0, 2)));}
%Y A360158 Cf. A002117, A013661, A016742, A016754, A056624, A344299, A350388, A360157.
%K A360158 nonn,easy,mult
%O A360158 1,9
%A A360158 _Amiram Eldar_, Jan 29 2023