A097974 Sum of distinct prime divisors of n which are <= sqrt(n).

0, 0, 0, 2, 0, 2, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 5, 2, 3, 2, 0, 10, 0, 2, 3, 2, 5, 5, 0, 2, 3, 7, 0, 5, 0, 2, 8, 2, 0, 5, 7, 7, 3, 2, 0, 5, 5, 9, 3, 2, 0, 10, 0, 2, 10, 2, 5, 5, 0, 2, 3, 14, 0, 5, 0, 2, 8, 2, 7, 5, 0, 7, 3, 2, 0, 12, 5, 2, 3, 2, 0, 10, 7, 2, 3, 2, 5, 5, 0, 9, 3, 7, 0, 5, 0

Offset: 1

Author

Leroy Quet, Sep 07 2004

Keywords

• nonn

Examples

			2 and 3 are the distinct prime divisors of 12 and both 2 and 3 are <= square root of 12. So a(12) = 2 + 3 = 5.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A027748, A063962.

Programs

Haskell

a097974 n = sum [p | p <- a027748_row n, p ^ 2 <= n]
-- Reinhard Zumkeller, Apr 05 2012

Maple

with(numtheory): a:=proc(n) local s,F,f,i: s:=0: F:=factorset(n): f:=nops(F): for i from 1 to f do if F[i]^2<=n then s:=s+F[i] else s:=s: fi od: s; end: seq(a(n),n=1..110); # Emeric Deutsch, Jan 30 2006

Mathematica

Do[Print[Plus @@ Select[Select[Divisors[n], PrimeQ], #<=Sqrt[n] &]], {n, 1, 100}] (* Ryan Propper, Jul 23 2005 *)
Table[DivisorSum[n, # &, And[PrimeQ@ #, # <= Sqrt[n]] &], {n, 103}] (* Michael De Vlieger, Sep 04 2017 *)

PARI

a(n) = sumdiv(n, d, d*isprime(d)*(d <= sqrt(n))); \\ Michel Marcus, Aug 17 2017

Formula

G.f.: Sum_{k>=1} prime(k) * x^(prime(k)^2) / (1 - x^prime(k)). - Ilya Gutkovskiy, Apr 04 2020

Extensions

More terms from Ryan Propper, Jul 23 2005

Further terms from Emeric Deutsch, Jan 30 2006