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

Author

Olivier Gérard

Keywords

• nonn

Links

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

Crossrefs

Cf. A001221, A023888.

Programs

Mathematica

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

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

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