A333750 Number of prime power divisors of n that are <= sqrt(n).

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 2, 0, 3, 0, 2, 1, 1, 1, 3, 0, 1, 1, 3, 0, 2, 0, 2, 2, 1, 0, 3, 1, 2, 1, 2, 0, 2, 1, 3, 1, 1, 0, 4, 0, 1, 2, 3, 1, 2, 0, 2, 1, 3, 0, 4, 0, 1, 2, 2, 1, 2, 0, 4, 2, 1, 0, 4, 1, 1, 1, 3, 0, 4, 1, 2, 1, 1, 1, 4, 0, 2, 2, 3

Offset: 1

Author

Ilya Gutkovskiy, Apr 03 2020

Keywords

• nonn

Links

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

Crossrefs

Cf. A001222, A038548, A063962, A069288, A069291, A073093, A246655, A333748, A333749, A333753.

Programs

Maple

f:= proc(n) local p;
  add(min(padic:-ordp(n,p),floor(1/2*log[p](n))),p=numtheory:-factorset(n))
end proc:
map(f, [$1..200]); # Robert Israel, Apr 22 2020

Mathematica

Table[DivisorSum[n, 1 &, # <= Sqrt[n] && PrimePowerQ[#] &], {n, 1, 100}]
nmax = 100; CoefficientList[Series[Sum[Boole[PrimePowerQ[k]] x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PARI

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

Formula

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