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A380322 a(n) is the sum of exponentially odd divisors of n^2.

%I A380322 #10 Jan 21 2025 13:33:24
%S A380322 1,3,4,11,6,12,8,43,31,18,12,44,14,24,24,171,18,93,20,66,32,36,24,172,
%T A380322 131,42,274,88,30,72,32,683,48,54,48,341,38,60,56,258,42,96,44,132,
%U A380322 186,72,48,684,351,393,72,154,54,822,72,344,80,90,60,264,62,96,248
%N A380322 a(n) is the sum of exponentially odd divisors of n^2.
%C A380322 The number of exponentially odd divisors of n^2 is equal to the number of divisors of n, A000005(n).
%H A380322 Amiram Eldar, <a href="/A380322/b380322.txt">Table of n, a(n) for n = 1..10000</a>
%F A380322 a(n) = A033634(A000290(n)) = A033634(n^2).
%F A380322 a(n) >= A000203(n), with equality if and only if n is squarefree (A005117).
%F A380322 Multiplicative with a(p^e) = (p^(2*e+1) - p)/(p^2 - 1) + 1.
%F A380322 Dirichlet g.f.: zeta(s-2) * zeta(s) * Product_{p prime} (1 - 1/p^(s-2) + 1/p^(s-1)).
%t A380322 f[p_, e_] := (p^(2*e+1) - p)/(p^2 - 1) + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A380322 (PARI) a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; (p^(2*e+1) - p)/(p^2 - 1) + 1);}
%Y A380322 Cf. A000005, A000203, A000290, A005117, A033634, A268335.
%Y A380322 Cf. A202535.
%K A380322 nonn,easy,mult
%O A380322 1,2
%A A380322 _Amiram Eldar_, Jan 20 2025