A070549 a(n) = Cardinality{ k in range 1 <= k <= n such that Moebius(k) = -1 }.

0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 20, 21, 22, 22, 22, 23, 24, 24, 25, 25, 25, 25, 25, 26

Offset: 1

Author

Benoit Cloitre, May 02 2002

Keywords

• easy
• nonn

Comments

mu(k)=-1 if k is the product of an odd number of distinct primes. See A057627 for mu(k)=0.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A008683, A057627.

Partial sums of A252233.

Cf. A002321, A013928, A030059, A070548, A104141.

Programs

Maple

ListTools:-PartialSums([seq(-min(numtheory:-mobius(n),0),n=1..100)]); # Robert Israel, Jan 08 2018

Mathematica

a[n_]:=Sum[Boole[MoebiusMu[k]==-1],{k,n}]; Array[a,78] (* Stefano Spezia, Jan 30 2023 *)

PARI

for(n=1,150,print1(sum(i=1,n,if(moebius(i)+1,0,1)),","))

Formula

From Amiram Eldar, Oct 01 2023: (Start)

a(n) = (A013928(n+1) - A002321(n))/2.

a(n) = A013928(n+1) - A070548(n).

a(n) = A070548(n) - A002321(n).

a(n) ~ (3/Pi^2) * n. (End)