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

1,10

Links

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

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)

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 *)

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, A324966, A345375.

Sequence in context: A328556 A321888 A321750 * A056929 A304096 A151692

Adjacent sequences: A345371 A345372 A345373 * A345375 A345376 A345377

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jun 16 2021

Status

approved