A345374 Number of unitary prime divisors of n whose prime index is odd.

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

Offset: 1

Author

Wesley Ivan Hurt, Jun 16 2021

Keywords

• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000720, A005117, A056169, A324966, A345375.

Programs

Mathematica

f[p_, e_] := If[e == 1 && OddQ[PrimePi[p]], 1, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 06 2023 *)

PARI

a(n) = {my(f = factor(n)); sum(i = 1, #f~, if(f[i,2] == 1 && primepi(f[i,1])%2, 1, 0));} \\ Amiram Eldar, Oct 06 2023

Formula

a(n) = Sum_{p|n, p prime} (pi(p) mod 2) * floor(1/gcd(p,n/p)).

From Amiram Eldar, Oct 06 2023: (Start)

a(n) = A056169(n) - A345375(n).

a(n) <= A324966(n), with equality if and only if n is squarefree (A005117).

Additive with a(p^e) = 1 if e = 1 and primepi(p) is odd and 0 otherwise. (End)