A228483 Duplicate of A129979.

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

Author

Wesley Ivan Hurt, Aug 22 2013

Keywords

• dead

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.

Is this a duplicate of A129979? - R. J. Mathar, Jul 18 2025

Examples

			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.
		

Links

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

Crossrefs

Cf. A001414, A008472, A008683.

Programs

Magma

[2-MoebiusMu(n): n in [1..100]]; // Vincenzo Librandi, Aug 23 2013

Maple

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

Mathematica

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

PARI

a(n) = 2 - moebius(n); \\ Michel Marcus, Apr 26 2016

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)).