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A365337 The sum of divisors of the largest exponentially odd number dividing n.

%I A365337 #8 Sep 02 2023 04:09:56
%S A365337 1,3,4,3,6,12,8,15,4,18,12,12,14,24,24,15,18,12,20,18,32,36,24,60,6,
%T A365337 42,40,24,30,72,32,63,48,54,48,12,38,60,56,90,42,96,44,36,24,72,48,60,
%U A365337 8,18,72,42,54,120,72,120,80,90,60,72,62,96,32,63,84,144,68
%N A365337 The sum of divisors of the largest exponentially odd number dividing n.
%C A365337 The number of divisors of the largest exponentially odd number dividing n is A286324(n).
%H A365337 Amiram Eldar, <a href="/A365337/b365337.txt">Table of n, a(n) for n = 1..10000</a>
%F A365337 a(n) = A000203(A350390(n)).
%F A365337 Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e is odd and (p^e-1)/(p-1) if e is even.
%F A365337 Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-2) + 1/p^(3*s-2)).
%F A365337 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p*(p^2-1))) = 1.2312911488886... (A065487). - _Amiram Eldar_, Sep 01 2023
%t A365337 f[p_, e_] := (p^(e + Mod[e, 2]) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365337 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(f[i,2] + f[i,2]%2) - 1)/(f[i,1] - 1));}
%Y A365337 Cf. A000203, A065487, A286324, A350390.
%K A365337 nonn,easy,mult
%O A365337 1,2
%A A365337 _Amiram Eldar_, Sep 01 2023