A351417 Number of divisors of n that are either prime or have at least one square divisor > 1.

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, 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, 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

Offset

1,4

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} [[Omega(d) = 1] = mu(d)^2], where [ ] is the Iverson bracket.

a(n) = A048105(n) + A001221(n). - Amiram Eldar, Oct 06 2023

Example

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).

Mathematica

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 *)

Prog

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

Crossrefs

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

Sequence in context: A069157 A294894 A076526 * A226378 A347460 A033273

Adjacent sequences: A351414 A351415 A351416 * A351418 A351419 A351420

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Feb 10 2022

Status

approved