A357698 a(n) is the sum of the aliquot divisors of n that are cubefree.

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 7, 1, 21, 1, 22, 11, 14, 1, 28, 6, 16, 13, 28, 1, 42, 1, 7, 15, 20, 13, 55, 1, 22, 17, 42, 1, 54, 1, 40, 33, 26, 1, 28, 8, 43, 21, 46, 1, 39, 17, 56, 23, 32, 1, 108, 1, 34, 41, 7, 19, 78, 1, 58, 27, 74, 1, 91, 1, 40

Offset: 1

Author

Amiram Eldar, Oct 10 2022

Keywords

• nonn

Examples

			The divisors of 16 that are cubefree are {1, 2, 4}, and their sum is a(16) = 1 + 2 + 4 = 7.
		

Links

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

Crossrefs

Cf. A004709, A073185, A212793, A293228.

Cf. A013661, A002117.

Programs

Mathematica

f[p_, e_] := 1 + p + If[e == 1, 0, p^2]; a[1] = 0; a[n_] := Times @@ f @@@ (fct = FactorInteger[n]) - If[AllTrue[fct[[;;, 2]], # < 3 &], n, 0]; Array[a, 100]

PARI

a(n) = {my(f = factor(n), s); s = prod(i=1, #f~, 1 + f[i,1] + if(f[i,2] == 1, 0, f[i,1]^2)); if(n==1 || vecmax(f[,2]) < 3, s -= n); s};

Formula

a(n) = Sum_{d|n, dA212793(d)*d.

a(n) = A073185(n) - (A212793(n)*n).

a(n) = 1 iff n is a prime.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2) - 1)/(2*zeta(3)) = 0.268262... .