A351414 Number of divisors of n that are either prime or have at least 1 square divisor > 1 and at least two distinct prime factors.

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 3, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, 1, 2, 3, 1, 3, 2, 3, 1, 7, 1, 2, 3, 3, 2, 3, 1, 5, 1, 2, 1, 6, 2, 2, 2, 4, 1, 6, 2, 3, 2, 2, 2, 6

Offset

1,6

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) + A361205(n). - Amiram Eldar, Oct 06 2023

Example

a(24) = 4; 24 has divisors 2,3 (primes) and 12,24 (which both have at least 1 square divisor > 1 and at least two distinct prime factors).
a(36) = 5; 36 has divisors 2,3 (primes) and 12,18,36 (which all have at least 1 square divisor > 1 and at least two distinct prime factors).

Mathematica

a[n_] := Module[{e = FactorInteger[n][[;; , 2]], d, nu, omega}, d = Times @@ (e+1); nu = Length[e]; omega = Total[e]; d - 2^nu - omega + 2*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), om = bigomega(f)); d - 2^nu - om + 2*nu; } \\ Amiram Eldar, Oct 06 2023

Crossrefs

Cf. A001221 (omega), A008683 (mu), A048105, A361205.

Sequence in context: A335450 A324191 A238946 * A349056 A326516 A081707

Adjacent sequences: A351411 A351412 A351413 * A351415 A351416 A351417

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Feb 10 2022

Status

approved