login
A105221
a(n) is the sum of n's distinct prime factors below n.
11
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
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
KEYWORD
easy,nonn
AUTHOR
Alexandre Wajnberg, Apr 13 2005
EXTENSIONS
Edited by Don Reble, Nov 17 2005
STATUS
approved