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Number of distinct prime divisors of n that are < sqrt(n).
38

%I #9 Apr 07 2020 13:27:36

%S 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 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 1,3,0,2,0,1,2,1,1,2,0,2,1,1,0,3,1,1,1,1,0,3

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

%C 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 Robert Israel, <a href="/A333806/b333806.txt">Table of n, a(n) for n = 1..10000</a>

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

%p N:= 100: # for a(1)..a(N)

%p V:= Vector(N):

%p p:= 1:

%p do

%p p:= nextprime(p);

%p if p^2 >= N then break fi;

%p L:= [seq(p*k,k=p+1..N/p)];

%p V[L]:= V[L]+~1

%p od:

%p convert(V,list); # _Robert Israel_, Apr 07 2020

%t Table[DivisorSum[n, 1 &, # < Sqrt[n] && PrimeQ[#] &], {n, 1, 90}]

%t nmax = 90; CoefficientList[Series[Sum[x^(Prime[k] (Prime[k] + 1))/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%o (PARI) a(n)=my(f=factor(n)[,1]); sum(i=1,#f, f[i]^2<n) \\ _Charles R Greathouse IV_, Apr 07 2020

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

%K nonn

%O 1,12

%A _Ilya Gutkovskiy_, Apr 05 2020