A333806 Number of distinct prime divisors of n that are < sqrt(n).

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

Offset: 1

Author

Ilya Gutkovskiy, Apr 05 2020

Keywords

• nonn

Comments

a(n) = 0 if and only if n = p^k where p is prime and k is 0, 1, or 2. - Charles R Greathouse IV, Apr 07 2020

Links

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

Crossrefs

Cf. A001221, A056924, A063962, A333805, A333808.

Programs

Maple

N:= 100: # for a(1)..a(N)
V:= Vector(N):
p:= 1:
do
  p:= nextprime(p);
  if p^2 >= N then break fi;
  L:= [seq(p*k,k=p+1..N/p)];
  V[L]:= V[L]+~1
od:
convert(V,list); # Robert Israel, Apr 07 2020

Mathematica

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

PARI

a(n)=my(f=factor(n)[,1]); sum(i=1,#f, f[i]^2Charles R Greathouse IV, Apr 07 2020

Formula

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

a(k*n) >= a(n) for k > 0. a(n^e) = A001221(n) for e > 2. - Richard Peterson, Dec 19 2024