A351394 - OEIS

%I #18 Oct 06 2023 10:54:11

%S 1,2,2,3,2,4,2,4,3,4,2,5,2,4,4,5,2,5,2,5,4,4,2,6,3,4,4,5,2,8,2,6,4,4,

%T 4,6,2,4,4,6,2,8,2,5,5,4,2,7,3,5,4,5,2,6,4,6,4,4,2,9,2,4,5,7,4,8,2,5,

%U 4,8,2,7,2,4,5,5,4,8,2,7,5,4,2,9,4,4,4,6,2,9,4,5,4,4,4

%N Number of divisors of n that are either squarefree, prime powers, or both.

%H Amiram Eldar, <a href="/A351394/b351394.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} sign(mu(d)^2 + [omega(d) = 1]).

%F a(n) = Sum_{d|n} (mu(d)^2 + [omega(d) = 1]*(1 - mu(d)^2)).

%F a(n) = A048105(n) + A046660(n). - _Amiram Eldar_, Oct 06 2023

%e a(36) = 6; 36 has 4 squarefree divisors 1,2,3,6 (where the primes 2 and 3 are both squarefree and 1st powers of primes) and 2 (additional) divisors that are powers of primes, 2^2 and 3^2.

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

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

%Y Cf. A001221, A008683, A046660, A048105, A246655.

%Y Cf. Similar to A327527.

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Feb 09 2022