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A374537 a(n) is the sum of the squares of the divisors of n that are exponentially odd numbers.

%I A374537 #10 Jul 12 2024 07:52:49
%S A374537 1,5,10,5,26,50,50,69,10,130,122,50,170,250,260,69,290,50,362,130,500,
%T A374537 610,530,690,26,850,739,250,842,1300,962,1093,1220,1450,1300,50,1370,
%U A374537 1810,1700,1794,1682,2500,1850,610,260,2650,2210,690,50,130,2900,850,2810
%N A374537 a(n) is the sum of the squares of the divisors of n that are exponentially odd numbers.
%C A374537 The number of divisors of n that are exponentially odd is A322483(n) and their sum is A033634(n).
%H A374537 Amiram Eldar, <a href="/A374537/b374537.txt">Table of n, a(n) for n = 1..10000</a>
%F A374537 a(n) = A001157(n) if and only if n is squarefree (A005117).
%F A374537 Multiplicative with a(p^e) = 1 + p^2 * (p^(4*floor((e-1)/2)+4) - 1) / (p^4 - 1).
%F A374537 Dirichlet g.f.: zeta(s) * zeta(2*s-4) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-4)).
%F A374537 Sum_{k=1..n} a(k) = c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 2/p^2 + 1/p^3) = A183699 * A065464 = 0.84677961058798544766... .
%t A374537 f[p_, e_] := 1 + p^2 * (p^(4*Floor[(e-1)/2]+4) - 1) / (p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A374537 (PARI) a(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); prod(i = 1, #p, 1 + p[i]^2 * (p[i]^(4*((e[i]-1)\2)+4) - 1) / (p[i]^4 - 1));}
%Y A374537 Cf. A001157, A005117, A033634, A322483, A351265, A358346, A374458, A374538.
%Y A374537 Cf. A002117, A013661, A065464, A183699.
%K A374537 nonn,easy,mult
%O A374537 1,2
%A A374537 _Amiram Eldar_, Jul 11 2024