cp's OEIS Frontend

A364988 a(n) is the sum of coreful divisors d of n such that n/d is also a coreful divisor.

%I A364988 #11 Aug 15 2023 07:44:05
%S A364988 1,0,0,2,0,0,0,6,3,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,5,0,12,0,0,0,0,30,0,
%T A364988 0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,62,0,0,0,
%U A364988 0,0,0,0,18,0,0,0,0,0,0,0,0,39,0,0,0,0
%N A364988 a(n) is the sum of coreful divisors d of n such that n/d is also a coreful divisor.
%C A364988 A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n (see A307958).
%C A364988 The number of these divisors is A361430(n).
%H A364988 Amiram Eldar, <a href="/A364988/b364988.txt">Table of n, a(n) for n = 1..10000</a>
%F A364988 Multiplicative with a(p^e) = (p^e - 1)/(p-1) - 1.
%F A364988 Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 + (2*p - p^s*(p+1))/p^(2*s)).
%F A364988 a(n) > 0 if and only if n is powerful (A001694).
%F A364988 a(n) <= n with equality only when n = 1.
%F A364988 a(p^2) = p for a prime p.
%t A364988 f[p_, e_] := (p^e - 1)/(p-1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A364988 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^f[i,2] - 1)/(f[i,1] - 1) - 1);}
%Y A364988 Cf. A001694, A307958, A361430.
%Y A364988 Similar sequences: A000203, A057723 (sum of coreful divisors).
%K A364988 nonn,mult
%O A364988 1,4
%A A364988 _Amiram Eldar_, Aug 15 2023