A368038 The sum of non-unitary divisors of the nonsquarefree numbers.

2, 6, 3, 8, 14, 9, 12, 24, 5, 12, 16, 30, 41, 36, 24, 18, 56, 7, 15, 28, 36, 48, 48, 24, 62, 36, 105, 20, 40, 84, 39, 64, 72, 54, 48, 120, 21, 36, 87, 84, 140, 112, 60, 42, 144, 11, 64, 30, 72, 126, 96, 72, 108, 96, 233, 28, 76, 60, 120, 54, 112, 180, 117, 84

Offset: 1

Author

Amiram Eldar, Dec 09 2023

Keywords

• nonn

Comments

The positive terms of A048146, since A048146(k) = 0 if and only if k is squarefree (A005117).

Links

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

Crossrefs

Cf. A005117, A048146, A013929.

Cf. A084936, A174961, A275699, A368039, A368040.

Cf. A002117, A013661, A072691, A229099.

Programs

Mathematica

f[p_, e_] := (p^(e+1)-1)/(p-1); nusigma[n_] := Module[{fct = FactorInteger[n]}, If[n == 1, 0, Times @@ f @@@ fct - Times @@ (1 + Power @@@ fct)]]; Select[Array[nusigma, 200], # > 0 &]

PARI

lista(kmax) = {my(f); for(k = 1, kmax, f = factor(k); if(!issquarefree(f), print1(sigma(f) - prod(i=1, #f~, 1+f[i,1]^f[i,2]), ", ")));}

Formula

a(n) = A048146(A013929(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)/2)*(1-1/zeta(3))/(1-1/zeta(2))^2 = 0.899359898779... .