A324966 - OEIS

%I #21 Oct 06 2023 04:55:44

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

%T 1,1,0,1,0,2,1,1,0,2,1,2,1,1,0,2,1,1,0,1,2,1,0,1,1,2,0,2,0,1,1,2,1,2,

%U 1,2,0,1,1,1,1,1,1,1,0,2,0,2,1,1,2,1,0

%N Number of distinct odd prime indices of n.

%C 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.

%C If x and y are coprime then a(x*y) = a(x)+a(y). - _Robert Israel_, Mar 24 2019

%H Robert Israel, <a href="/A324966/b324966.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A001221(n) - A324967(n). - _Robert Israel_, Mar 24 2019

%F G.f.: Sum_{k>=1} x^prime(2*k-1) / (1 - x^prime(2*k-1)). - _Ilya Gutkovskiy_, Feb 12 2020

%F Additive with a(p^e) = 1 if primepi(p) is odd and 0 otherwise. - _Amiram Eldar_, Oct 06 2023

%e 180180 has prime indices {1,1,2,2,3,4,5,6}, so a(180180) = 3.

%p f:= proc(n) nops(select(type,map(numtheory:-pi,numtheory:-factorset(n)),odd)) end proc:

%p map(f, [$1..100]); # _Robert Israel_, Mar 24 2019

%t Table[Count[If[n==1,{},FactorInteger[n]],{_?(OddQ[PrimePi[#]]&),_}],{n,100}]

%o (PARI) a(n) = my(f=factor(n)[,1]); sum(k=1, #f, primepi(f[k]) % 2); \\ _Michel Marcus_, Mar 22 2019

%Y Cf. A000720, A001221, A003963, A005087, A066208, A112798, A257991, A257992, A289509, A324929, A324967.

%K nonn,easy

%O 1,10

%A _Gus Wiseman_, Mar 21 2019