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

1,10

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

Links

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

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

Example

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

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}]

Prog

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

Crossrefs

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

Sequence in context: A364249 A191904 A265892 * A005090 A073490 A279907

Adjacent sequences: A324963 A324964 A324965 * A324967 A324968 A324969

Keyword

nonn,easy

Author

Gus Wiseman, Mar 21 2019

Status

approved