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A351417
Number of divisors of n that are either prime or have at least one square divisor > 1.
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
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
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Feb 10 2022
STATUS
approved