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A369758 The sum of divisors of the smallest cubefull exponentially odd number that is divisible by n.

%I A369758 #13 Feb 03 2024 10:22:14
%S A369758 1,15,40,15,156,600,400,15,40,2340,1464,600,2380,6000,6240,63,5220,
%T A369758 600,7240,2340,16000,21960,12720,600,156,35700,40,6000,25260,93600,
%U A369758 30784,63,58560,78300,62400,600,52060,108600,95200,2340,70644,240000,81400,21960,6240
%N A369758 The sum of divisors of the smallest cubefull exponentially odd number that is divisible by n.
%C A369758 First differs from A369720 at n = 16.
%H A369758 Amiram Eldar, <a href="/A369758/b369758.txt">Table of n, a(n) for n = 1..10000</a>
%F A369758 a(n) = A000203(A356192(n)).
%F A369758 Multiplicative with a(p) = p^3 + p^2 + p + 1, a(p^e) = (p^(e+1)-1)/(p-1) for an odd e >= 3, and a(p^e) = (p^(e+2)-1)/(p-1) for an even e.
%F A369758 a(n) >= A000203(n), with equality if and only if n is cubefull exponentially odd number (A335988).
%F A369758 Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-3) + 1/p^(s-2) + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(3*s-5) - 1/p^(3*s-4) - 1/p^(3*s-3) + 1/p^(4*s-5) + 1/p^(4*s-4)).
%F A369758 Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(4) * zeta(6) * Product_{p prime} (1 - 1/p^4 - 1/p^6 + 1/p^10 + 1/p^11 - 1/p^13) = 1.00040193512214077945... .
%F A369758 Equivalently, c = Product_{p prime} (1 + 1/(p^3*(p^4 - 1)*(p^4 + p^2 + 1))). - _Vaclav Kotesovec_, Feb 02 2024
%t A369758 f[p_, e_] := (p^If[OddQ[e], Max[e, 3] + 1, e + 2] - 1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A369758 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^if(f[i,2]%2, max(f[i,2], 3) + 1, f[i,2] + 2) - 1)/(f[i,1] - 1));}
%o A369758 (Python)
%o A369758 from math import prod
%o A369758 from sympy import factorint
%o A369758 def A369758(n): return prod((p**((3 if e==1 else e)+1+(e&1^1))-1)//(p-1) for p,e in factorint(n).items()) # _Chai Wah Wu_, Feb 03 2024
%Y A369758 Cf. A000203, A335988, A356192, A365349, A369720, A369757, A369759.
%Y A369758 Cf. A013662, A013664.
%K A369758 nonn,easy,mult
%O A369758 1,2
%A A369758 _Amiram Eldar_, Jan 31 2024