cp's OEIS Frontend

A367991 The sum of the divisors of the squarefree part of n.

%I A367991 #13 Jun 03 2025 11:26:08
%S A367991 1,3,4,1,6,12,8,3,1,18,12,4,14,24,24,1,18,3,20,6,32,36,24,12,1,42,4,8,
%T A367991 30,72,32,3,48,54,48,1,38,60,56,18,42,96,44,12,6,72,48,4,1,3,72,14,54,
%U A367991 12,72,24,80,90,60,24,62,96,8,1,84,144,68,18,96,144,72
%N A367991 The sum of the divisors of the squarefree part of n.
%C A367991 First differs from A348503 at n = 72 and from A344695 at n = 108.
%C A367991 The sum of the infinitary divisors (A077609) of n that are squarefree (A005117). - _Amiram Eldar_, Jun 03 2025
%H A367991 Amiram Eldar, <a href="/A367991/b367991.txt">Table of n, a(n) for n = 1..10000</a>
%F A367991 Multiplicative with a(p^e) = p + 1 if e is odd and 1 otherwise.
%F A367991 a(n) = A000203(A007913(n)) = A048250(A007913(n)).
%F A367991 a(n) = A048250(n)/A367990(n).
%F A367991 a(n) >= 1, with equality if and only if n is a square (A000290).
%F A367991 a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).
%F A367991 Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^s).
%F A367991 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4)/zeta(3) = 0.900392677639... .
%t A367991 f[p_, e_] := If[OddQ[e], p + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A367991 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, f[i,1]+1, 1));}
%Y A367991 Cf. A000203, A000290, A005117, A007913, A048250, A077609, A367990.
%Y A367991 Cf. A002117, A013662.
%Y A367991 Cf. A344695, A348503.
%K A367991 nonn,easy,mult
%O A367991 1,2
%A A367991 _Amiram Eldar_, Dec 07 2023