A131088 -id:A131088 - cp's OEIS frontend

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

Gary W. Adamson, Jun 14 2007

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)

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

Cf. A131088, A051731, A054525, A007427.

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

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

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

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

More terms from Michel Marcus, Feb 26 2022

A131089 a(n) = Sum_{d|n} (2 - mu(d)).

Original entry on oeis.org

1, 4, 4, 6, 4, 8, 4, 8, 6, 8, 4, 12, 4, 8, 8, 10, 4, 12, 4, 12, 8, 8, 4, 16, 6, 8, 8, 12, 4, 16, 4, 12, 8, 8, 8, 18, 4, 8, 8, 16, 4, 16, 4, 12, 12, 8, 4, 20, 6, 12, 8, 12, 4, 16, 8, 16, 8, 8, 4, 24, 4, 8, 12, 14, 8, 16, 4, 12, 8, 16, 4, 24, 4, 8, 12, 12, 8, 16

Offset: 1

Gary W. Adamson, Jun 14 2007

nonn

Apart from the first term the same as A062011. - Andrew Howroyd, Aug 09 2018

			a(4) = 6 = (2 + 3 + 0 + 1), row 4 of A131088.

Row sums of triangle A131088.

Cf. A062011, A051731, A131090, A228483.

[1] cat [2*NumberOfDivisors(n) : n in [2..100] ]; // Vincenzo Librandi, Aug 10 2018

Join[{1}, Table[2 DivisorSigma[0, n], {n, 100}]] (* Vincenzo Librandi, Aug 10 2018 *)

a(n)={2*numdiv(n) - (n==1)} \\ Andrew Howroyd, Aug 09 2018

a(n) = 2*tau(n) = A062011(n) for n > 1. - Andrew Howroyd, Aug 09 2018

Inverse Moebius transform of A228483.

Name changed and terms a(10) and beyond from Andrew Howroyd, Aug 09 2018

cp's OEIS Frontend

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

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