A169646 Number of squarefree numbers of form k*n, 1 <= k <= n.

1, 1, 2, 0, 3, 2, 5, 0, 0, 3, 7, 0, 8, 5, 6, 0, 11, 0, 12, 0, 8, 9, 15, 0, 0, 10, 0, 0, 17, 8, 19, 0, 13, 13, 15, 0, 23, 15, 17, 0, 26, 11, 28, 0, 0, 18, 30, 0, 0, 0, 21, 0, 32, 0, 25, 0, 23, 23, 36, 0, 37, 25, 0, 0, 30, 18, 41, 0, 29, 22, 44, 0, 45, 30, 0, 0, 36, 22, 49, 0, 0, 32, 51, 0, 41, 34

Offset: 1

Author

Reinhard Zumkeller, Apr 05 2010

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..3000

Crossrefs

Cf. A000040, A008966, A065473, A073311, A112929, A118259.

Programs

Magma

[&+[MoebiusMu(k*n)^2: k in [1..n]]: n in [1..80]]; // Vincenzo Librandi, Jul 25 2019

Maple

with(numtheory): seq(add(mobius(n*i)^2, i = 1 .. n), n = 1 .. 90); # Ridouane Oudra, Jul 24 2019

Mathematica

Count[#,?SquareFreeQ]&/@Table[k*n,{n,90},{k,n}] (* _Harvey P. Dale, Sep 05 2012 *)
Table[Sum[MoebiusMu[i n]^2, {i, n}], {n, 100}] (* Vincenzo Librandi, Jul 25 2019 *)

PARI

a(n) = sum(i = 1, n, issquarefree(i*n)); \\ Amiram Eldar, May 12 2025

Formula

a(n) = A008966(n)*A073311(n).

a(A000040(n)) = A112929(n).

a(n) = Sum_{i=1..n} A008966(n*i). - Ridouane Oudra, Jul 24 2019

a(n) = (A118259(n) - A118259(n-1))/2, for n>1. - Ridouane Oudra, May 04 2025

Sum_{k=1..n} a(k) ~ c * n / 2, where c = Product_{p prime} (1 - (3*p-2)/(p^3)) (A065473). - Amiram Eldar, May 12 2025