A345375 Number of unitary prime divisors of n whose prime index is even.

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

Offset

1,21

Links

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

Formula

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

From Amiram Eldar, Oct 06 2023: (Start)

a(n) = A056169(n) - A345374(n).

a(n) <= A324967(n), with equality if and only if n is squarefree (A005117).

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A000720, A005117, A056169, A324967, A345374.

Sequence in context: A025462 A024879 A024316 * A101669 A243067 A366092

Adjacent sequences: A345372 A345373 A345374 * A345376 A345377 A345378

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jun 16 2021

Status

approved