A333753 Sum of prime power divisors of n that are <= sqrt(n).

0, 0, 0, 2, 0, 2, 0, 2, 3, 2, 0, 5, 0, 2, 3, 6, 0, 5, 0, 6, 3, 2, 0, 9, 5, 2, 3, 6, 0, 10, 0, 6, 3, 2, 5, 9, 0, 2, 3, 11, 0, 5, 0, 6, 8, 2, 0, 9, 7, 7, 3, 6, 0, 5, 5, 13, 3, 2, 0, 14, 0, 2, 10, 14, 5, 5, 0, 6, 3, 14, 0, 17, 0, 2, 8, 6, 7, 5, 0, 19, 12, 2, 0, 16, 5, 2, 3, 14, 0, 19

Offset: 1

Author

Ilya Gutkovskiy, Apr 03 2020

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A023889, A066839, A069289, A069293, A097974, A246655, A333750, A333751, A333752.

Programs

Maple

f:= proc(n) local F,i,j,t;
  F:= ifactors(n)[2];
  t:= 0;
  for i from 1 to nops(F) do
    j:= min(F[i,2],ilog[F[i,1]^2](n));
    t:= t + (F[i,1]^j-1)*F[i,1]/(F[i,1]-1)
  od;
  t
end proc:
map(f, [$1..100]); # Robert Israel, Feb 15 2023

Mathematica

Table[DivisorSum[n, # &, # <= Sqrt[n] && PrimePowerQ[#] &], {n, 1, 90}]
nmax = 90; CoefficientList[Series[Sum[Boole[PrimePowerQ[k]] k x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PARI

a(n) = sumdiv(n, d, if ((d^2<=n) && isprimepower(d), d)); \\ Michel Marcus, Apr 03 2020

Formula

G.f.: Sum_{p prime, k>=1} p^k * x^(p^(2*k)) / (1 - x^(p^k)).