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A360163 a(n) is the sum of the square roots of the divisors of n that are odd squares.

%I A360163 #8 Jan 29 2023 12:05:44
%S A360163 1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,6,1,4,1,1,1,1,1,1,1,
%T A360163 1,4,1,1,1,1,1,1,1,1,4,1,1,1,8,6,1,1,1,4,1,1,1,1,1,1,1,1,4,1,1,1,1,1,
%U A360163 1,1,1,4,1,1,6,1,1,1,1,1,13,1,1,1,1,1,1
%N A360163 a(n) is the sum of the square roots of the divisors of n that are odd squares.
%C A360163 First differs from A336649 at n = 27.
%H A360163 Amiram Eldar, <a href="/A360163/b360163.txt">Table of n, a(n) for n = 1..10000</a>
%F A360163 a(n) = Sum_{d|n, d odd square} sqrt(d).
%F A360163 a(n) = (A069290(n) + A347176(n))/2.
%F A360163 a(n) = A069290(n) if n is not a multiple of 4.
%F A360163 Multiplicative with a(2^e) = 1, and a(p^e) = (p^(floor(e/2)+1)-1)/(p-1) for p > 2.
%F A360163 Dirichlet g.f.: zeta(s)*zeta(2*s-1)*(1-2^(1-2*s)).
%F A360163 Sum_{k=1..n} a(k) ~ (n/4) * (log(n) + 3*gamma - 1 + 2*log(2)), where gamma is Euler's constant (A001620).
%t A360163 f[p_, e_] := (p^(Floor[e/2] + 1) - 1)/(p - 1); f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100]
%o A360163 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, 1, (f[i, 1]^(floor(f[i, 2]/2)+1) - 1)/(f[i, 1] - 1))); }
%Y A360163 Cf. A001620, A069290, A336649.
%K A360163 nonn,easy,mult
%O A360163 1,9
%A A360163 _Amiram Eldar_, Jan 29 2023