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

1,4

Links

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

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

Example

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

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, #<k&], {k, 86}] (* Giorgos Kalogeropoulos, Nov 21 2021 *)

Prog

(Haskell)
a105221 n = a008472 n - n * fromIntegral (a010051 n)
-- Reinhard Zumkeller, Apr 05 2013
(PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, if (f[k, 1]<n, 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

Crossrefs

Cf. A003508, A008472, A010051.

Sequence in context: A171388 A086134 A071090 * A215339 A061376 A058974

Adjacent sequences: A105218 A105219 A105220 * A105222 A105223 A105224

Keyword

easy,nonn

Author

Alexandre Wajnberg, Apr 13 2005

Extensions

Edited by Don Reble, Nov 17 2005

Status

approved