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A360159 a(n) is the sum of divisors of n that are odd squares.

%I A360159 #8 Jan 29 2023 12:14:26
%S A360159 1,1,1,1,1,1,1,1,10,1,1,1,1,1,1,1,1,10,1,1,1,1,1,1,26,1,10,1,1,1,1,1,
%T A360159 1,1,1,10,1,1,1,1,1,1,1,1,10,1,1,1,50,26,1,1,1,10,1,1,1,1,1,1,1,1,10,
%U A360159 1,1,1,1,1,1,1,1,10,1,1,26,1,1,1,1,1,91,1,1
%N A360159 a(n) is the sum of divisors of n that are odd squares.
%H A360159 Amiram Eldar, <a href="/A360159/b360159.txt">Table of n, a(n) for n = 1..10000</a>
%F A360159 a(n) = Sum_{d|n, d odd square} d.
%F A360159 a(n) = (A035316(n) + A344300(n))/2.
%F A360159 Multiplicative with a(2^e) = 1, and for p > 2, a(p^e) = (p^(e+2)-1)/(p^2-1) for even e and a(p^e) = (p^(e+1)-1)/(p^2-1) for odd e.
%F A360159 Dirichlet g.f.: zeta(s)*zeta(2s-2)*(1-4^(1-s)).
%F A360159 Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)/6 = 0.4353958914... .
%t A360159 f[p_, e_] := (p^(2*(1 + Floor[e/2])) - 1)/(p^2 - 1); f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A360159 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, 1, (f[i, 1]^(2*(f[i, 2]\2)+2)-1)/(f[i, 1]^2-1))); }
%Y A360159 Cf. A016754, A035316, A078434, A344300.
%K A360159 nonn,easy,mult
%O A360159 1,9
%A A360159 _Amiram Eldar_, Jan 29 2023