A105221 a(n) is the sum of n's distinct prime factors below n.

0, 0, 0, 2, 0, 5, 0, 2, 3, 7, 0, 5, 0, 9, 8, 2, 0, 5, 0, 7, 10, 13, 0, 5, 5, 15, 3, 9, 0, 10, 0, 2, 14, 19, 12, 5, 0, 21, 16, 7, 0, 12, 0, 13, 8, 25, 0, 5, 7, 7, 20, 15, 0, 5, 16, 9, 22, 31, 0, 10, 0, 33, 10, 2, 18, 16, 0, 19, 26, 14, 0, 5, 0, 39, 8, 21, 18, 18, 0, 7, 3, 43, 0, 12, 22, 45

Offset: 1

Author

Alexandre Wajnberg, Apr 13 2005

Keywords

• easy
• nonn

Examples

			a(12)=5 because 12's distinct prime factors 2 and 3 sum to 5.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..1000

Crossrefs

Cf. A003508, A008472, A010051.

Programs

Haskell

a105221 n = a008472 n - n * fromIntegral (a010051 n)
-- Reinhard Zumkeller, Apr 05 2013

Maple

f:= n -> convert(numtheory:-factorset(n) minus {n}, `+`):
map(f, [$1..100]); # Robert Israel, Sep 18 2023

Mathematica

Table[Total@Select[Join@@Union@*Table@@@FactorInteger@k,#Giorgos Kalogeropoulos, Nov 21 2021 *)

PARI

a(n) = my(f=factor(n)); sum(k=1, #f~, if (f[k,1]Michel Marcus, Nov 21 2021

Python

from sympy import primefactors
def A105221(n): return sum(p for p in primefactors(n) if p < n) # Chai Wah Wu, Sep 18 2023

Formula

a(n) = A008472(n) - A010051(n) * n. - Reinhard Zumkeller, Apr 05 2013

G.f.: Sum_{k>=1} prime(k) * x^(2*prime(k)) / (1 - x^prime(k)). - Ilya Gutkovskiy, Apr 13 2021

Extensions

Edited by Don Reble, Nov 17 2005