cp's OEIS Frontend

A384554 The sum of the infinitary divisors of n that are cubefree.

%I A384554 #13 Jun 03 2025 15:16:42
%S A384554 1,3,4,5,6,12,8,7,10,18,12,20,14,24,24,1,18,30,20,30,32,36,24,28,26,
%T A384554 42,13,40,30,72,32,3,48,54,48,50,38,60,56,42,42,96,44,60,60,72,48,4,
%U A384554 50,78,72,70,54,39,72,56,80,90,60,120,62,96,80,5,84,144,68,90
%N A384554 The sum of the infinitary divisors of n that are cubefree.
%C A384554 The number of these divisors is A368883(n), and the largest of them is A384555(n).
%C A384554 The sum of the infinitary divisors of n that are squarefree (A005117) is A367991(n).
%H A384554 Amiram Eldar, <a href="/A384554/b384554.txt">Table of n, a(n) for n = 1..10000</a>
%F A384554 Multiplicative with a(p^e) = 1 if e == 0 (mod 4), p + 1 if e == 1 (mod 4), p^2 + 1 if e == 2 (mod 4), and p^2 + p + 1 if e == 3 (mod 4).
%F A384554 a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).
%F A384554 a(n) <= A049417(n), with equality if and only if n is cubefree (A004709).
%F A384554 Dirichlet g.f.: zeta(4*s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^s + 1/p^(2*s-2) + 1/p^(2*s) + 1/p^(3*s-1) + 1/p^(3*s-2) + 1/p^(3*s)).
%F A384554 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(8) * Product_{p prime} (1 + 1/p^2 - 2/p^3 + 2/p^4 - 1/p^5 - 1/p^7) = 1.2351002232125595782019... .
%t A384554 f[p_, e_] := Switch[Mod[e, 4], 0, 1, 1, p+1, 2, p^2+1, 3, p^2+p+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A384554 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; [1, p+1, p^2+1, p^2+p+1][e%4+1]);}
%o A384554 (Python)
%o A384554 from math import prod
%o A384554 from sympy import factorint
%o A384554 def A384554(n): return prod((1,p+1,p**2+1,p*(p+1)+1)[e&3] for p,e in factorint(n).items()) # _Chai Wah Wu_, Jun 03 2025
%Y A384554 Cf. A000203, A005117, A004709, A013666, A049417, A073185, A077609, A367991, A368883, A384555.
%K A384554 nonn,easy,mult
%O A384554 1,2
%A A384554 _Amiram Eldar_, Jun 03 2025