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

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, 34, 96, 15, 38, 70, 39, 42, 66, 30, 46, 90, 14, 33, 80, 78, 126, 98, 58, 39, 90, 11, 62, 30, 42, 126, 66, 60, 102, 70, 216, 21, 74, 30, 114, 51, 78, 150, 78, 82, 126, 13

Offset

1,1

Comments

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

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.

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

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

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

Mathematica

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 &]

Prog

(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, ", "))); }

Crossrefs

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

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

Sequence in context: A280342 A275476 A185380 * A136695 A337115 A354372

Adjacent sequences: A373055 A373056 A373057 * A373059 A373060 A373061

Keyword

nonn,easy

Author

Amiram Eldar, May 21 2024

Status

approved