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A167515 The sum over the divisors of n, except the maximum-prime-power divisors collected in A008475.

%I A167515 #20 May 17 2019 02:36:35
%S A167515 1,1,1,3,1,7,1,7,4,11,1,21,1,15,16,15,1,28,1,33,22,23,1,49,6,27,13,45,
%T A167515 1,62,1,31,34,35,36,78,1,39,40,77,1,84,1,69,64,47,1,105,8,66,52,81,1,
%U A167515 91,56,105,58,59,1,156,1,63,88,63,66,128,1,105,70
%N A167515 The sum over the divisors of n, except the maximum-prime-power divisors collected in A008475.
%C A167515 If n = Product (p_j^k_j) is the standard prime power decomposition of n, there is a set of size A001221(n) which contains the divisors which are largest powers of primes, {p_1^k_1, p_2^k_2, ..., p_j^k_j}. a(n) sums all the divisors not in this set. If p, q, ..., z are distinct primes, k are natural numbers (A000027), p^k prime powers (A000961), the following formulas hold: a(p) = 1. a(pq) = pq+1. a(pq...z) = (p+1)* (q+1)* ... *(z+1) - (p+q+ ...+z). a(p^k) = (p^k-1)/(p-1).
%H A167515 Nathaniel Johnston, <a href="/A167515/b167515.txt">Table of n, a(n) for n = 1..10000</a>
%F A167515 a(n) = A000203(n) - A008475(n).
%F A167515 a(n) = A178636(n) + 1.
%e A167515 For n = 12, set of prime-power-factor divisors of 12: {3, 4}, set of non-(prime-power-factor) divisors on 12: {1, 2, 6, 12}. a(12) = 1+2+6+12=21.
%p A167515 A008475 := proc(n) add( op(1,d)^op(2,d),d= ifactors(n)[2] ) ; end proc:
%p A167515 A167515 := proc(n) numtheory[sigma](n)-A008475(n) ; end proc:
%p A167515 seq(A167515(n),n=1..80) ; # _R. J. Mathar_, Dec 21 2010
%Y A167515 Cf. A000203, A008475, A178636.
%K A167515 nonn
%O A167515 1,4
%A A167515 _Jaroslav Krizek_, Dec 15 2010