A178636 -id:A178636 - cp's OEIS frontend

A159077 a(n) = A008475(n) + 1.

1, 3, 4, 5, 6, 6, 8, 9, 10, 8, 12, 8, 14, 10, 9, 17, 18, 12, 20, 10, 11, 14, 24, 12, 26, 16, 28, 12, 30, 11, 32, 33, 15, 20, 13, 14, 38, 22, 17, 14, 42, 13, 44, 16, 15, 26, 48, 20, 50, 28, 21, 18, 54, 30, 17, 16, 23, 32, 60, 13, 62, 34, 17, 65, 19, 17, 68, 22, 27, 15, 72, 18, 74

Offset: 1

Jaroslav Krizek, Apr 04 2009

nonn

If n = Product (p_i^k_i) for i = 1, …, j then a(n) is sum of divisor d from set of divisors{1, p_1^k_1, p_2^k_2, …, p_j^k_j}.

			For n = 12, set of divisors {1, p_1^k_1, p_2^k_2, …, p_j^k_j}: {1, 3, 4}. a(12) = 1+3+4=8.

Cf. A008475, A023888.

f[n_] := 1 + Plus @@ Power @@@ FactorInteger@ n; f[1] = 1; Array[f, 60]

a(n)=local(t); if(n<1, 0, t=factor(n); 1+sum(k=1, matsize(t)[1], t[k, 1]^t[k, 2])) /* Anton Mosunov, Jan 05 2017 */

a(n) = [Sum_(i=1,…, j) p_i^k_i] + 1 = A000203(n) - A178636(n).

a(1) = 1, a(p) = p+1, a(pq) = p+q+1, a(pq...z) = p+q+...+z+1, a(p^k) = p^k+1, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.

Edited by N. J. A. Sloane, Apr 07 2009

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

Original entry on oeis.org

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

Jaroslav Krizek, Dec 15 2010

nonn

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).

			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.

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

Cf. A000203, A008475, A178636.

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

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

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

cp's OEIS Frontend

A159077 a(n) = A008475(n) + 1.

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