cp's OEIS Frontend

A360165 a(n) is the sum of the square roots of the unitary divisors of n that are odd squares minus the sum of the square roots of the unitary divisors of n that are even squares.

%I A360165 #10 Jan 29 2023 12:08:13
%S A360165 1,1,1,-1,1,1,1,1,4,1,1,-1,1,1,1,-3,1,4,1,-1,1,1,1,1,6,1,1,-1,1,1,1,1,
%T A360165 1,1,1,-4,1,1,1,1,1,1,1,-1,4,1,1,-3,8,6,1,-1,1,1,1,1,1,1,1,-1,1,1,4,
%U A360165 -7,1,1,1,-1,1,1,1,4,1,1,6,-1,1,1,1,-3,10,1
%N A360165 a(n) is the sum of the square roots of the unitary divisors of n that are odd squares minus the sum of the square roots of the unitary divisors of n that are even squares.
%C A360165 The unitary analog of A347176.
%H A360165 Amiram Eldar, <a href="/A360165/b360165.txt">Table of n, a(n) for n = 1..10000</a>
%F A360165 a(n) = Sum_{d|n, gcd(d, n/d)=1, d odd square} (-1)^(d+1)*sqrt(d).
%F A360165 a(n) = A360164(n) - 2 * A360162(n).
%F A360165 Multiplicative with a(2^e) = 1 - 2^(e/2) if e is even and 1 otherwise, and for p > 2, a(p^e) = p^(e/2) + 1 if e is even and 1 if e is odd.
%F A360165 Dirichlet g.f.: (zeta(s)*zeta(2*s-1)/zeta(3*s-1))*(2^(3*s)-2^(s+2)+2)/(2^(3*s)-2).
%F A360165 Sum_{k=1..n} a(k) ~ (n/Pi^2)*(log(n) + 3*gamma - 1 + 4*log(2) - 3*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).
%t A360165 f[p_, e_] := If[OddQ[e], 1, p^(e/2) + 1]; f[2, e_] := If[OddQ[e], 1, 1 - 2^(e/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A360165 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, if(f[i, 2]%2, 1, 1 - f[i, 1]^(f[i, 2]/2)), if(f[i, 2]%2, 1, f[i, 1]^(f[i, 2]/2) + 1))); }
%Y A360165 Cf. A001620, A306016, A347176, A360162, A360164.
%K A360165 sign,easy,mult
%O A360165 1,9
%A A360165 _Amiram Eldar_, Jan 29 2023