cp's OEIS Frontend

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

%I A178637 #16 Jan 30 2021 21:38:11
%S A178637 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,
%T A178637 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,
%U A178637 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
%N A178637 a(n) = sum of divisors d of n such that d is not equal to p^k where p = prime, k >=1.
%H A178637 Antti Karttunen, <a href="/A178637/b178637.txt">Table of n, a(n) for n = 1..65537</a>
%F A178637 a(n) = A000203(n) - A023889(n) = A035321(n) + 1.
%F A178637 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.
%F A178637 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
%e A178637 For n = 12, set of such divisors is {1, 6, 12}; a(12) = 1+6+12 = 19.
%t A178637 Array[Plus @@ (Select[Divisors[#], (Length[FactorInteger[#]] > 1) &]) &, 100] + 1 (* _Robert P. P. McKone_, Jan 28 2021 *)
%o A178637 (PARI) A178637(n) = sumdiv(n,d,(omega(d)!=1)*(d)); \\ _Antti Karttunen_, Aug 06 2018
%Y A178637 One more than A035321.
%Y A178637 Cf. A000203, A001221 (omega), A023889, A035321.
%K A178637 nonn
%O A178637 1,6
%A A178637 _Jaroslav Krizek_, Dec 25 2010