A351394 Number of divisors of n that are either squarefree, prime powers, or both.

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, 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, 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

Offset

1,2

Links

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

Index entries for sequences computed from exponents in factorization of n

Formula

a(n) = Sum_{d|n} sign(mu(d)^2 + [omega(d) = 1]).

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

a(n) = A048105(n) + A046660(n). - Amiram Eldar, Oct 06 2023

Example

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.

Mathematica

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 *)

Prog

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

Crossrefs

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

Cf. Similar to A327527.

Sequence in context: A289872 A301855 A080256 * A327527 A337454 A289849

Adjacent sequences: A351391 A351392 A351393 * A351395 A351396 A351397

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Feb 09 2022

Status

approved