%I #31 Sep 08 2022 08:46:05

%S 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,

%T 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,

%U 1,3,3,2,3,1,2,2,1,3,3,2,2,1,3,2,1,1,1

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

%C 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.

%C 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.

%H Vincenzo Librandi, <a href="/A228483/b228483.txt">Table of n, a(n) for n = 1..1000</a>

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

%F 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)).

%e a(19) = 3 because mu(19) = -1 and 2 - (-1) = 3.

%e a(20) = 2 because mu(20) = 0 and 2 - 0 = 2.

%e a(21) = 1 because mu(21) = 1 and 2 - 1 = 1.

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

%t 2 - MoebiusMu[Range[100]] (* _Alonso del Arte_, Aug 22 2013 *)

%o (Magma) [2-MoebiusMu(n): n in [1..100]]; // _Vincenzo Librandi_, Aug 23 2013

%o (PARI) a(n) = 2 - moebius(n); \\ _Michel Marcus_, Apr 26 2016

%Y Cf. A001414, A008472, A008683.

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Aug 22 2013