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A373058 The sum of the aliquot coreful divisors of the nonsquarefree numbers.

%I A373058 #13 Jul 23 2024 10:51:15
%S A373058 2,6,3,6,14,6,10,18,5,12,14,30,36,30,22,15,42,7,10,26,24,42,30,21,62,
%T A373058 34,96,15,38,70,39,42,66,30,46,90,14,33,80,78,126,98,58,39,90,11,62,
%U A373058 30,42,126,66,60,102,70,216,21,74,30,114,51,78,150,78,82,126,13
%N A373058 The sum of the aliquot coreful divisors of the nonsquarefree numbers.
%C A373058 A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958).
%C A373058 The positive terms of A336563: if k is a squarefree number (A005117) then the only coreful divisor of k is k itself, so k has no aliquot coreful divisors.
%C A373058 The number of the aliquot coreful divisors of the n-th nonsquarefree number is A368039(n).
%H A373058 Amiram Eldar, <a href="/A373058/b373058.txt">Table of n, a(n) for n = 1..10000</a>
%F A373058 a(n) = A336563(A013929(n)).
%F A373058 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (A065487 - 1)/(1-1/zeta(2))^2 = 1.50461493205911656114... .
%t A373058 f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; Select[Array[s, 300], # > 0 &]
%o A373058 (PARI) lista(kmax) = {my(f); for(k = 1, kmax, f = factor(k); if(!issquarefree(f), print1(prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1) - 1)/(f[i, 1] - 1) - 1) - k, ", "))); }
%o A373058 (Python)
%o A373058 from math import prod, isqrt
%o A373058 from sympy import mobius, factorint
%o A373058 def A373058(n):
%o A373058     def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A373058     m, k = n, f(n)
%o A373058     while m != k:
%o A373058         m, k = k, f(k)
%o A373058     return prod((p**(e+1)-1)//(p-1)-1 for p, e in factorint(m).items())-m # _Chai Wah Wu_, Jul 22 2024
%Y A373058 Cf. A005117, A013661, A013929, A065487, A229099, A307958, A336563.
%Y A373058 Cf. A084936, A174961, A275699, A368038, A368039, A368040, A368541, A368713.
%K A373058 nonn,easy
%O A373058 1,1
%A A373058 _Amiram Eldar_, May 21 2024