A228483 a(n) = 2 - mu(n), where mu(n) is the Moebius function (A008683).

1, 3, 3, 2, 3, 1, 3, 2, 2, 1, 3, 2, 3, 1, 1, 2, 3, 2, 3, 2, 1, 1, 3, 2, 2, 1, 2, 2, 3, 3, 3, 2, 1, 1, 1, 2, 3, 1, 1, 2, 3, 3, 3, 2, 2, 1, 3, 2, 2, 2, 1, 2, 3, 2, 1, 2, 1, 1, 3, 2, 3, 1, 2, 2, 1, 3, 3, 2, 1, 3, 3, 2, 3, 1, 2, 2, 1, 3, 3, 2, 2, 1, 3, 2, 1, 1, 1

Offset

1,2

Comments

1 <= a(n) <= 3: a(n) = 1 when n is both squarefree and has an even number of distinct prime factors (or if n = 1). So a(n) = 1 when mu(n) = 1. a(n) = 2 when n is square-full. a(n) = 3 when n is both squarefree and has an odd number of distinct prime factors.

When n is semiprime, a(n) is equal to the ratio of the number of prime factors of n (with multiplicity) to the number of its distinct prime factors. Analogously, when n is semiprime, a(n) is equal to the ratio of the sum of the prime factors of n (with repetition) to the sum of its distinct prime factors.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Formula

a(n) = 2 - mu(n) = 2 - A008683(n).

a(A001358(n)) = 5 - tau(A001358(n)) = 3 - omega(A001358(n)) = 3 + 2*A001358(n) - sigma(A001358(n)) - phi(A001358(n)) = Omega(A001358(n))/omega(A001358(n))= sopfr(A001358(n))/sopf(A001358(n)).

Example

a(19) = 3 because mu(19) = -1 and 2 - (-1) = 3.
a(20) = 2 because mu(20) = 0 and 2 - 0 = 2.
a(21) = 1 because mu(21) = 1 and 2 - 1 = 1.

Maple

with(numtheory); seq(2-mobius(k), k=1..70);

Mathematica

2 - MoebiusMu[Range[100]] (* Alonso del Arte, Aug 22 2013 *)

Prog

(Magma) [2-MoebiusMu(n): n in [1..100]]; // Vincenzo Librandi, Aug 23 2013
(PARI) a(n) = 2 - moebius(n); \\ Michel Marcus, Apr 26 2016

Crossrefs

Cf. A001414, A008472, A008683.

Sequence in context: A014967 A210851 A120992 * A129979 A274709 A260896

Adjacent sequences: A228480 A228481 A228482 * A228484 A228485 A228486

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 22 2013

Status

approved