A351418 Number of divisors of n that are either of the form p^k (p prime, k>1) or are nonprime squarefree numbers.

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 5, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, 6, 2, 5, 1, 3, 2, 5, 1, 5, 1, 2, 3, 3, 2, 5, 1, 5, 4, 2, 1, 6, 2, 2, 2, 4, 1, 6, 2, 3, 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] = (1-mu(d)^2)], where [ ] is the Iverson bracket.

a(n) = A034444(n) - 2*A001221(n) + A001222(n). - Amiram Eldar, Oct 06 2023

Example

a(60) = 6; 30 has the divisors 1,6,10,15,30 (nonprime squarefree numbers), and 4 = 2^2 (which is of the form p^k, k>1).

Mathematica

a[n_] := Module[{e = FactorInteger[n][[;; , 2]], nu}, nu = Length[e]; 2^nu - 2*nu + Total[e]]; 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 - 2*nu + om; } \\ Amiram Eldar, Oct 06 2023

Crossrefs

Cf. A001221 (omega), A001222 (bigomega), A008683 (mu), A034444.

Sequence in context: A086436 A001222 A257091 * A359909 A319269 A320888

Adjacent sequences: A351415 A351416 A351417 * A351419 A351420 A351421

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Feb 10 2022

Status

approved