A351417 - OEIS

%I #13 Oct 06 2023 10:54:45

%S 0,1,1,2,1,2,1,3,2,2,1,4,1,2,2,4,1,4,1,4,2,2,1,6,2,2,3,4,1,3,1,5,2,2,

%T 2,7,1,2,2,6,1,3,1,4,4,2,1,8,2,4,2,4,1,6,2,6,2,2,1,7,1,2,4,6,2,3,1,4,

%U 2,3,1,10,1,2,4,4,2,3,1,8,4,2,1,7,2,2,2,6,1,7,2,4,2,2,2

%N Number of divisors of n that are either prime or have at least one square divisor > 1.

%H Amiram Eldar, <a href="/A351417/b351417.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(n) = Sum_{d|n} [[Omega(d) = 1] = mu(d)^2], where [ ] is the Iverson bracket.

%F a(n) = A048105(n) + A001221(n). - _Amiram Eldar_, Oct 06 2023

%e a(96) = 10; 96 has divisors 2,3 (prime) and 4,8,12,16,24,32,48,96 (all with at least one square divisor > 1).

%t a[n_] := Module[{e = FactorInteger[n][[;;, 2]], nu}, nu = Length[e]; Times @@ (e+1) - 2^nu + nu]; a[1] = 0; Array[a, 100] (* _Amiram Eldar_, Oct 06 2023 *)

%o (PARI) a(n) = {my(f = factor(n), d = numdiv(f), nu = omega(f)); d - 2^nu + nu;} \\ _Amiram Eldar_, Oct 06 2023

%Y Cf. A001221 (omega), A001222 (Omega, aka bigomega), A008683 (mu), A048105.

%K nonn,easy

%O 1,4

%A _Wesley Ivan Hurt_, Feb 10 2022