cp's OEIS Frontend

A369720 The sum of divisors of the smallest cubefull number that is a multiple of n.

%I A369720 #7 Jan 30 2024 04:43:00
%S A369720 1,15,40,15,156,600,400,15,40,2340,1464,600,2380,6000,6240,31,5220,
%T A369720 600,7240,2340,16000,21960,12720,600,156,35700,40,6000,25260,93600,
%U A369720 30784,63,58560,78300,62400,600,52060,108600,95200,2340,70644,240000,81400,21960,6240
%N A369720 The sum of divisors of the smallest cubefull number that is a multiple of n.
%H A369720 Amiram Eldar, <a href="/A369720/b369720.txt">Table of n, a(n) for n = 1..10000</a>
%F A369720 a(n) = A000203(A356193(n)).
%F A369720 Multiplicative with a(p) = p^3 + p^2 + p + 1 for e <= 2, and a(p^e) = (p^(e+1)-1)/(p-1) for e >= 3.
%F A369720 a(n) >= A000203(n), with equality if and only if n is cubefull (A036966).
%F A369720 Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-3) + 1/p^(s-2) - 1/p^(2*s-4) - 1/p^(2*s-3) - 1/p^(2*s-2) + 1/p^(4*s-4)).
%F A369720 Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(3) * zeta(4) * Product_{p prime} (1 - 1/p^3 - 1/p^4 + 1/p^7 + 1/p^12 - 1/p^13) = 1.00015013207437782094... .
%t A369720 f[p_, e_] := (p^If[e <= 2, 4, e + 1]-1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]
%o A369720 (PARI) a(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i,2] <= 2, f[i,2] = 3)); sigma(f);}
%Y A369720 Cf. A000203, A036966, A356193, A369717, A369719, A369721.
%Y A369720 Cf. A002117, A013662, A183700.
%K A369720 nonn,easy,mult
%O A369720 1,2
%A A369720 _Amiram Eldar_, Jan 30 2024