A077641 Number of squarefree integers in closed interval [n, 2n-1], i.e., among n consecutive numbers beginning with n.

1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 7, 8, 8, 8, 8, 9, 10, 11, 12, 13, 14, 15, 14, 15, 15, 16, 16, 17, 18, 19, 19, 19, 20, 21, 21, 22, 23, 23, 23, 24, 24, 25, 25, 26, 27, 28, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 37, 38, 37, 38, 38, 39, 38, 39, 40, 41, 41, 41, 42, 43, 43, 44, 45, 45

Offset: 1

Author

Labos Elemer, Nov 14 2002

Keywords

• nonn

Examples

			For n = 10: among the numbers {10,...,19} seven are squarefree: {10,11,13,14,15,17,19}, so a(10) = 7.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005117, A008683, A008966, A013661.

Programs

Mathematica

Table[Apply[Plus, Table[Abs[MoebiusMu[w+j]], {j, 0, w-1}]], {w, 1, 128}]
Table[Count[Range[n,2n-1],?SquareFreeQ],{n,80}] (* _Harvey P. Dale, Oct 27 2013 *)
Module[{nn=80,sf},sf=Table[If[SquareFreeQ[n],1,0],{n,2nn}];Table[Total[ Take[ sf,{i,2i-1}]],{i,nn}]] (* Harvey P. Dale, May 20 2016 *)

PARI

a(n) = sum(i = 0, n-1, issquarefree(n+i)); \\ Amiram Eldar, Feb 25 2025

Formula

a(n) = Sum_{j=0..n-1} abs(mu(n+j)).

a(1) = 1; a(n + 1) = a(n) - issquarefree(n) + issquarefree(2n-2) + issquarefree(2n-1) for n > 0. - David A. Corneth, May 20 2016

a(n) ~ n/zeta(2). - Amiram Eldar, Feb 25 2025