A129979 a(n) = 2-mu(n), where mu=A008683 is the Moebius function.

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, 2, 3, 2

Offset: 1

Author

Gary W. Adamson, Jun 14 2007

Keywords

• nonn
• easy

Comments

Left border of the triangle A131088.

From Wesley Ivan Hurt, Aug 22 2013 (Start):

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.

(End)

Examples

			A131088 = (1; 3,1; 3,0,1; 2,3,0,1; ...).
		

Crossrefs

Cf. A131088, A051731, A054525, A007427.

Programs

Maple

with(numtheory); seq(2-mobius(k),k=1..70); # Wesley Ivan Hurt, Aug 22 2013

Mathematica

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

PARI

T(n,k) = 2*!(n%k) - if (!(n % k), moebius(n/k), 0); \\ A131088
a(n) =  T(n, 1); \\ Michel Marcus, Feb 26 2022

Formula

Inverse Moebius transform of A007427 with changed signs except for A007427(1) = 1; i.e., inverse Moebius transform of (1, 2, 2, -1, 2, -4, 2, 0, -1, -4, ...).

a(n) = 2 - mu(n) = 2 - A008683(n). - Wesley Ivan Hurt, Aug 22 2013

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))= A001414(A001358(n))/A008472(A001358(n)). - Wesley Ivan Hurt, Aug 22 2013

Extensions

More terms from Michel Marcus, Feb 26 2022