A183099 a(n) = sum of powerful divisors d (excluding 1) of n.

0, 0, 0, 4, 0, 0, 0, 12, 9, 0, 0, 4, 0, 0, 0, 28, 0, 9, 0, 4, 0, 0, 0, 12, 25, 0, 36, 4, 0, 0, 0, 60, 0, 0, 0, 49, 0, 0, 0, 12, 0, 0, 0, 4, 9, 0, 0, 28, 49, 25, 0, 4, 0, 36, 0, 12, 0, 0, 0, 4, 0, 0, 9, 124, 0, 0, 0, 4, 0, 0, 0, 129, 0, 0, 25, 4, 0, 0, 0, 28, 117, 0, 0, 4, 0, 0, 0, 12, 0, 9, 0, 4, 0, 0, 0, 60, 0, 49, 9, 129

Offset

1,4

Comments

a(n) = sum of divisors d of n from set A001694(m) - powerful numbers for m >=2.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16385

Index entries for sequences related to sums of divisors.

Formula

a(n) = A000203(n) - A183100(n) = A183097(n) - 1.

a(1) = 0, a(p) = 0, a(pq) = 0, a(pq...z) = 0, a(p^k) = ((p^(k+1)-1) / (p-1))-p-1, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.

Example

For n = 12, set of such divisors is {4}; a(12) = 4.

Mathematica

f[p_, e_] := (p^(e+1)-1)/(p-1) - p; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - 1; Array[a, 100] (* Amiram Eldar, Aug 29 2023 *)

Prog

(PARI) A183099(n) = (sumdiv(n, d, ispowerful(d)*d) - 1); \\ Antti Karttunen, Oct 07 2017

Crossrefs

Cf. A000203, A001694, A183097, A183100.

Sequence in context: A236379 A126849 A284117 * A162296 A364103 A169773

Adjacent sequences: A183096 A183097 A183098 * A183100 A183101 A183102

Keyword

nonn,easy

Author

Jaroslav Krizek, Dec 25 2010

Status

approved