A360159 - OEIS

%I #8 Jan 29 2023 12:14:26

%S 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 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 1,1,1,1,1,1,1,1,10,1,1,26,1,1,1,1,1,91,1,1

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

%H Amiram Eldar, <a href="/A360159/b360159.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n, d odd square} d.

%F a(n) = (A035316(n) + A344300(n))/2.

%F 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 Dirichlet g.f.: zeta(s)*zeta(2s-2)*(1-4^(1-s)).

%F Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)/6 = 0.4353958914... .

%t 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 (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 Cf. A016754, A035316, A078434, A344300.

%K nonn,easy,mult

%O 1,9

%A _Amiram Eldar_, Jan 29 2023