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

1,4

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).

Links

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

Formula

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

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

Example

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.

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

Crossrefs

Cf. A000203, A008475, A178636.

Sequence in context: A347963 A186428 A323599 * A140435 A194181 A277934

Adjacent sequences: A167512 A167513 A167514 * A167516 A167517 A167518

Keyword

nonn

Author

Jaroslav Krizek, Dec 15 2010

Status

approved