A345375 - OEIS

%I #9 Oct 06 2023 04:55:59

%S 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,

%T 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,

%U 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

%N Number of unitary prime divisors of n whose prime index is even.

%H Amiram Eldar, <a href="/A345375/b345375.txt">Table of n, a(n) for n = 1..10000</a>

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

%F From _Amiram Eldar_, Oct 06 2023: (Start)

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

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

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

%t 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 *)

%o (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

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

%K nonn,easy

%O 1,21

%A _Wesley Ivan Hurt_, Jun 16 2021