

A228483


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


3



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
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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 squarefull. 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(2mobius(k), k=1..70);


MATHEMATICA

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


PROG

(MAGMA) [2MoebiusMu(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: A210851 A120992 A129979 * A274709 A260896 A237347
Adjacent sequences: A228480 A228481 A228482 * A228484 A228485 A228486


KEYWORD

nonn,easy


AUTHOR

Wesley Ivan Hurt, Aug 22 2013


STATUS

approved



