A023889 Sum of the prime power divisors of n (not including 1).

0, 2, 3, 6, 5, 5, 7, 14, 12, 7, 11, 9, 13, 9, 8, 30, 17, 14, 19, 11, 10, 13, 23, 17, 30, 15, 39, 13, 29, 10, 31, 62, 14, 19, 12, 18, 37, 21, 16, 19, 41, 12, 43, 17, 17, 25, 47, 33, 56, 32, 20, 19, 53, 41, 16, 21, 22, 31, 59, 14, 61, 33, 19, 126, 18, 16, 67, 23, 26, 14

Offset

1,2

Links

Ivan Neretin, Table of n, a(n) for n = 1..10000

Formula

G.f.: Sum_{k>=2} floor(1/omega(k))*k*x^k/(1 - x^k), where omega(k) is the number of distinct prime factors (A001221). - Ilya Gutkovskiy, Jan 04 2017

a(n) = A023888(n) - 1. - Michel Marcus, Mar 21 2017

a(n) = Sum_{d|n} d * [omega(d) = 1], where omega is the number of distinct prime factors and [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 28 2021

Mathematica

Array[ Plus @@ (Select[ Divisors[ # ], PrimePowerQ ])&, 80 ]

Prog

(PARI) a(n) = sumdiv(n, d, if(isprimepower(d), d)); \\ Michel Marcus, Mar 21 2017; corrected by Daniel Suteu, Jul 20 2018
(PARI) a(n) = my(f = factor(n)); sum(k = 1, #f~, f[k, 1] * (f[k, 1]^f[k, 2] - 1) / (f[k, 1] - 1)) \\ Daniel Suteu, Jul 20 2018

Crossrefs

Cf. A001221, A023888.

Sequence in context: A137761 A100769 A353977 * A327669 A253413 A337355

Adjacent sequences: A023886 A023887 A023888 * A023890 A023891 A023892

Keyword

nonn

Author

Olivier Gérard

Status

approved