A324966 Number of distinct odd prime indices of n.

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

Offset: 1

Author

Gus Wiseman, Mar 21 2019

Keywords

• nonn
• easy

Comments

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.

If x and y are coprime then a(x*y) = a(x)+a(y). - Robert Israel, Mar 24 2019

Examples

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

Links

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

Crossrefs

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

Programs

Maple

f:= proc(n) nops(select(type,map(numtheory:-pi,numtheory:-factorset(n)),odd)) end proc:
map(f, [$1..100]); # Robert Israel, Mar 24 2019

Mathematica

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

PARI

a(n) = my(f=factor(n)[,1]); sum(k=1, #f, primepi(f[k]) % 2); \\ Michel Marcus, Mar 22 2019

Formula

a(n) = A001221(n) - A324967(n). - Robert Israel, Mar 24 2019

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

Additive with a(p^e) = 1 if primepi(p) is odd and 0 otherwise. - Amiram Eldar, Oct 06 2023