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

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, 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, 91, 56, 105, 58, 59, 1, 156, 1, 63, 88, 63, 66, 128, 1, 105, 70

Offset: 1

Author

Jaroslav Krizek, Dec 15 2010

Keywords

• nonn

Comments

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).

Examples

			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.
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A008475, A178636.

Programs

Maple

A008475 := proc(n) add( op(1,d)^op(2,d),d= ifactors(n)[2] ) ; end proc:
A167515 := proc(n) numtheory[sigma](n)-A008475(n) ; end proc:
seq(A167515(n),n=1..80) ; # R. J. Mathar, Dec 21 2010

Formula

a(n) = A000203(n) - A008475(n).

a(n) = A178636(n) + 1.