A324967 - OEIS

A324967

Number of distinct even prime indices of n.

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

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OFFSET

1,21

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) - A324966(n). - Robert Israel, Mar 24 2019

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

Additive with a(p^e) = 1 if primepi(p) is even 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)), even)) end proc:

map(f, [$1..100]); # Robert Israel, Mar 24 2019

MATHEMATICA

Table[Count[If[n==1, {}, FactorInteger[n]], {_?(EvenQ[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, A324966.

Sequence in context: A179317 A330657 A307598 * A320332 A358007 A348956

Adjacent sequences: A324964 A324965 A324966 * A324968 A324969 A324970

KEYWORD

nonn,easy

AUTHOR

Gus Wiseman, Mar 21 2019

STATUS

approved