A319072 a(n) is the sum of the non-bi-unitary divisors of n.

0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 8, 0, 0, 0, 4, 0, 9, 0, 12, 0, 0, 0, 0, 5, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, 0, 24, 18, 0, 0, 16, 7, 15, 0, 28, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 24, 8, 0, 0, 0, 36, 0, 0, 0, 45, 0, 0, 20, 40, 0, 0, 0, 24, 9, 0, 0, 64, 0, 0, 0, 0, 0, 54, 0, 48, 0, 0, 0, 0, 0, 21, 36, 87

Offset

1,4

Links

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

Formula

a(n) = A000203(n) - A188999(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = A072691 - A307160 = 0.069628292421818906692... . - Amiram Eldar, Nov 25 2025

Example

For n = 12 the divisors of 12 are 1, 2, 3, 4, 6, 12, and the bi-unitary divisors of 12 are 1, 3, 4, 12, hence the non-bi-unitary divisors of 12 are 2 and 6, and the sum of them is 2 + 6 = 8, so a(12) = 8. Also the sum of the divisors of 12 is 28, and the sum of the bi-unitary divisors of 12 is 20, so a(12) = 28 - 20 = 8.

Mathematica

f1[p_, e_] := (p^(e+1) - 1)/(p - 1); f2[p_, e_] := f1[p, e] - If[OddQ[e], 0, p^(e/2)]; a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; Array[a, 100] (* Amiram Eldar, Apr 04 2024 *)

Prog

(PARI) a(n) = {my(f = factor(n)); sigma(f) - prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1)-1)/(f[i, 1]-1) - if(!(f[i, 2] % 2), f[i, 1]^(f[i, 2]/2))); } \\ Amiram Eldar, Nov 25 2025

Crossrefs

Cf. A000203, A027750, A188999, A222266.

Cf. A072691, A307160.

Sequence in context: A218234 A092197 A213618 * A348271 A083804 A341840

Adjacent sequences: A319069 A319070 A319071 * A319073 A319074 A319075

Keyword

nonn,easy

Author

Omar E. Pol, Sep 22 2018

Status

approved