A023889 - OEIS

%I #25 Jan 30 2021 20:54:21

%S 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,

%T 39,13,29,10,31,62,14,19,12,18,37,21,16,19,41,12,43,17,17,25,47,33,56,

%U 32,20,19,53,41,16,21,22,31,59,14,61,33,19,126,18,16,67,23,26,14

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

%H Ivan Neretin, <a href="/A023889/b023889.txt">Table of n, a(n) for n = 1..10000</a>

%F 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

%F a(n) = A023888(n) - 1. - _Michel Marcus_, Mar 21 2017

%F 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

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

%o (PARI) a(n) = sumdiv(n, d, if(isprimepower(d), d)); \\ _Michel Marcus_, Mar 21 2017; corrected by _Daniel Suteu_, Jul 20 2018

%o (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

%Y Cf. A001221, A023888.

%K nonn

%O 1,2

%A _Olivier Gérard_