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

1,2

Comments

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}.

Links

Table of n, a(n) for n=1..73.

Formula

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.

Example

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.

Mathematica

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

Prog

(PARI) 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 */

Crossrefs

Cf. A008475, A023888.

Sequence in context: A036288 A320727 A374691 * A049267 A111608 A126800

Adjacent sequences: A159074 A159075 A159076 * A159078 A159079 A159080

Keyword

nonn

Author

Jaroslav Krizek, Apr 04 2009

Extensions

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

Status

approved