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

1,2

Links

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

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A005117, A008683, A008966, A013661.

Sequence in context: A030566 A007963 A137222 * A329547 A330560 A358466

Adjacent sequences: A077638 A077639 A077640 * A077642 A077643 A077644

Keyword

nonn

Author

Labos Elemer, Nov 14 2002

Status

approved