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A360161 a(n) is the sum of unitary divisors of n that are odd squares minus the sum of unitary divisors of n that are even squares.

%I A360161 #11 Jan 29 2023 12:12:27
%S A360161 1,1,1,-3,1,1,1,1,10,1,1,-3,1,1,1,-15,1,10,1,-3,1,1,1,1,26,1,1,-3,1,1,
%T A360161 1,1,1,1,1,-30,1,1,1,1,1,1,1,-3,10,1,1,-15,50,26,1,-3,1,1,1,1,1,1,1,
%U A360161 -3,1,1,10,-63,1,1,1,-3,1,1,1,10,1,1,26,-3,1,1,1
%N A360161 a(n) is the sum of unitary divisors of n that are odd squares minus the sum of unitary divisors of n that are even squares.
%C A360161 The unitary analog of A344300.
%H A360161 Amiram Eldar, <a href="/A360161/b360161.txt">Table of n, a(n) for n = 1..10000</a>
%F A360161 a(n) = Sum_{d|n, gcd(d, n/d)=1, d square} (-1)^(d+1) * d.
%F A360161 a(n) = A360160(n) - 2 * A358347(n).
%F A360161 Multiplicative with a(2^e) = 1 - 2^e if e is even and 1 if e is odd, and for p > 2, a(p^e) = p^e + 1 if e is even and 1 if e is odd.
%F A360161 Dirichlet g.f.: (zeta(s)*zeta(2*s-2)/zeta(3*s-2))*(2^(3*s)-2^(s+3)+4)/(2^(3*s)-4).
%F A360161 Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)/(3*zeta(5/2)*(4*sqrt(2)-1)) = 0.1393911255... .
%t A360161 f[p_, e_] := If[OddQ[e], 1, p^e + 1]; f[2, e_] := If[OddQ[e], 1, 1 - 2^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A360161 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, if(f[i, 2]%2, 1, 1 - 2^f[i, 2]), if(f[i, 2]%2, 1, f[i, 1]^f[i, 2] + 1))); }
%Y A360161 Cf. A016742, A016754, A078434, A247041, A344300, A358347, A360160.
%K A360161 sign,easy,mult
%O A360161 1,4
%A A360161 _Amiram Eldar_, Jan 29 2023