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A324852 Number of distinct prime indices of n that divide n.

%I A324852 #19 Jan 11 2025 03:11:32
%S A324852 0,1,0,1,0,2,0,1,0,1,0,2,0,1,1,1,0,2,0,1,0,1,0,2,0,1,0,2,0,3,0,1,0,1,
%T A324852 0,2,0,1,0,1,0,2,0,1,1,1,0,2,0,1,0,1,0,2,1,2,0,1,0,3,0,1,0,1,0,2,0,1,
%U A324852 0,1,0,2,0,1,1,1,0,3,0,1,0,1,0,3,0,1,0,1,0,3,0,1,0,1,0,2,0,1,0,1,0,2,0,1,1
%N A324852 Number of distinct prime indices of n that divide n.
%C A324852 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%H A324852 Alois P. Heinz, <a href="/A324852/b324852.txt">Table of n, a(n) for n = 1..65536</a>
%F A324852 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} 1/(k*prime(k)) = 0.848969... (A124012). - _Amiram Eldar_, Jan 11 2025
%e A324852 60060 has 7 prime indices {1,1,2,3,4,5,6}, all of which divide 60060, and 6 of which are distinct, so a(60060) = 6.
%p A324852 a:= n-> add(`if`(irem(n, numtheory[pi](i[1]))=0, 1, 0), i=ifactors(n)[2]):
%p A324852 seq(a(n), n=1..120);  # _Alois P. Heinz_, Mar 19 2019
%t A324852 Table[Count[If[n==1,{},FactorInteger[n]],{p_,_}/;Divisible[n,PrimePi[p]]],{n,100}]
%o A324852 (PARI) a(n) = {my(f = factor(n)[,1]); sum(k=1, #f, !(n % primepi(f[k])));} \\ _Michel Marcus_, Mar 19 2019
%Y A324852 The version for all prime indices (counted with multiplicity) is A324848.
%Y A324852 Positions of zeros are A324846.
%Y A324852 Positions of ones are A323440.
%Y A324852 Cf. A000720, A001222, A003963, A056239, A120383, A124012.
%Y A324852 Cf. A324704, A324771, A324847, A324849, A324850, A324853, A324856.
%K A324852 nonn
%O A324852 1,6
%A A324852 _Gus Wiseman_, Mar 18 2019