cp's OEIS Frontend

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

%I A333806 #23 Dec 20 2024 07:52:58
%S A333806 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,
%T A333806 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,
%U A333806 1,3,0,2,0,1,2,1,1,2,0,2,1,1,0,3,1,1,1,1,0,3
%N A333806 Number of distinct prime divisors of n that are < sqrt(n).
%C A333806 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
%H A333806 Robert Israel, <a href="/A333806/b333806.txt">Table of n, a(n) for n = 1..10000</a>
%F A333806 G.f.: Sum_{k>=1} x^(prime(k)*(prime(k) + 1)) / (1 - x^prime(k)).
%F A333806 a(k*n) >= a(n) for k > 0. a(n^e) = A001221(n) for e > 2. - _Richard Peterson_, Dec 19 2024
%p A333806 N:= 100: # for a(1)..a(N)
%p A333806 V:= Vector(N):
%p A333806 p:= 1:
%p A333806 do
%p A333806   p:= nextprime(p);
%p A333806   if p^2 >= N then break fi;
%p A333806   L:= [seq(p*k,k=p+1..N/p)];
%p A333806   V[L]:= V[L]+~1
%p A333806 od:
%p A333806 convert(V,list); # _Robert Israel_, Apr 07 2020
%t A333806 Table[DivisorSum[n, 1 &, # < Sqrt[n] && PrimeQ[#] &], {n, 1, 90}]
%t A333806 nmax = 90; CoefficientList[Series[Sum[x^(Prime[k] (Prime[k] + 1))/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%o A333806 (PARI) a(n)=my(f=factor(n)[,1]); sum(i=1,#f, f[i]^2<n) \\ _Charles R Greathouse IV_, Apr 07 2020
%Y A333806 Cf. A001221, A056924, A063962, A333805, A333808.
%K A333806 nonn
%O A333806 1,12
%A A333806 _Ilya Gutkovskiy_, Apr 05 2020