A178637 a(n) = sum of divisors d of n such that d is not equal to p^k where p = prime, k >=1.

1, 1, 1, 1, 1, 7, 1, 1, 1, 11, 1, 19, 1, 15, 16, 1, 1, 25, 1, 31, 22, 23, 1, 43, 1, 27, 1, 43, 1, 62, 1, 1, 34, 35, 36, 73, 1, 39, 40, 71, 1, 84, 1, 67, 61, 47, 1, 91, 1, 61, 52, 79, 1, 79, 56, 99, 58, 59, 1, 154, 1, 63, 85, 1, 66, 128, 1, 103, 70, 130, 1, 169, 1, 75, 91, 115, 78, 150, 1, 151, 1, 83, 1, 208, 86, 87, 88, 155, 1, 215, 92, 139, 94, 95, 96, 187, 1, 113, 133, 181

Offset: 1

Author

Jaroslav Krizek, Dec 25 2010

Keywords

• nonn

Examples

			For n = 12, set of such divisors is {1, 6, 12}; a(12) = 1+6+12 = 19.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

One more than A035321.

Cf. A000203, A001221 (omega), A023889, A035321.

Programs

Mathematica

Array[Plus @@ (Select[Divisors[#], (Length[FactorInteger[#]] > 1) &]) &, 100] + 1 (* Robert P. P. McKone, Jan 28 2021 *)

PARI

A178637(n) = sumdiv(n,d,(omega(d)!=1)*(d)); \\ Antti Karttunen, Aug 06 2018

Formula

a(n) = A000203(n) - A023889(n) = A035321(n) + 1.

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

a(n) = Sum_{d|n} d * (1 - [omega(n) = 1]), where omega is the number of distinct prime factors (A001221) and [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 28 2021