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 A167515 The sum over the divisors of n, except the maximum-prime-power divisors collected in A008475. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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: A316553 A186428 A323599 * A140435 A194181 A277934 Adjacent sequences:  A167512 A167513 A167514 * A167516 A167517 A167518 KEYWORD nonn AUTHOR Jaroslav Krizek, Dec 15 2010 STATUS approved

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Last modified May 16 02:39 EDT 2021. Contains 343937 sequences. (Running on oeis4.)