A097974 - OEIS

%I #31 Apr 04 2020 21:25:12

%S 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,

%T 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,

%U 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

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

%H Reinhard Zumkeller, <a href="/A097974/b097974.txt">Table of n, a(n) for n = 1..10000</a>

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

%e 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.

%p 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

%t Do[Print[Plus @@ Select[Select[Divisors[n], PrimeQ], #<=Sqrt[n] &]], {n, 1, 100}] (* _Ryan Propper_, Jul 23 2005 *)

%t Table[DivisorSum[n, # &, And[PrimeQ@ #, # <= Sqrt[n]] &], {n, 103}] (* _Michael De Vlieger_, Sep 04 2017 *)

%o (Haskell)

%o a097974 n = sum [p | p <- a027748_row n, p ^ 2 <= n]

%o -- _Reinhard Zumkeller_, Apr 05 2012

%o (PARI) a(n) = sumdiv(n, d, d*isprime(d)*(d <= sqrt(n))); \\ _Michel Marcus_, Aug 17 2017

%Y Cf. A027748, A063962.

%K nonn

%O 1,4

%A _Leroy Quet_, Sep 07 2004

%E More terms from _Ryan Propper_, Jul 23 2005

%E Further terms from _Emeric Deutsch_, Jan 30 2006