A373058 - OEIS

%I #13 Jul 23 2024 10:51:15

%S 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 34,96,15,38,70,39,42,66,30,46,90,14,33,80,78,126,98,58,39,90,11,62,

%U 30,42,126,66,60,102,70,216,21,74,30,114,51,78,150,78,82,126,13

%N The sum of the aliquot coreful divisors of the nonsquarefree numbers.

%C A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958).

%C 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 The number of the aliquot coreful divisors of the n-th nonsquarefree number is A368039(n).

%H Amiram Eldar, <a href="/A373058/b373058.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A336563(A013929(n)).

%F Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (A065487 - 1)/(1-1/zeta(2))^2 = 1.50461493205911656114... .

%t 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 (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 (Python)

%o from math import prod, isqrt

%o from sympy import mobius, factorint

%o def A373058(n):

%o     def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))

%o     m, k = n, f(n)

%o     while m != k:

%o         m, k = k, f(k)

%o     return prod((p**(e+1)-1)//(p-1)-1 for p, e in factorint(m).items())-m # _Chai Wah Wu_, Jul 22 2024

%Y Cf. A005117, A013661, A013929, A065487, A229099, A307958, A336563.

%Y Cf. A084936, A174961, A275699, A368038, A368039, A368040, A368541, A368713.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, May 21 2024