A077381 Number of squarefree numbers between successive squares (exclusive).

2, 3, 5, 5, 7, 8, 8, 11, 11, 14, 14, 14, 17, 19, 18, 20, 22, 20, 24, 26, 28, 26, 28, 30, 31, 32, 33, 36, 34, 37, 40, 36, 43, 42, 44, 46, 47, 46, 49, 48, 48, 51, 50, 56, 55, 57, 58, 60, 63, 59, 63, 63, 63, 69, 70, 67, 71, 71, 73, 71, 74, 78, 76, 78, 81, 79, 84, 83, 87, 85, 84, 87

Offset

1,1

Links

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Gabriel Mincu and Laurenţiu Panaitopol, On some properties of squarefree and squareful numbers, Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie, Nouvelle Série, Vol. 49 (97), No. 1 (2006), pp. 63-68; alternative link.

Formula

From Amiram Eldar, Feb 16 2021: (Start)

a(n) > n for all n (Mincu and Panaitopol, 2006).

a(n) ~ (12/Pi^2) * n. (End)

Example

a(1) = 2 because there are 2 squarefree integers between 1^2 and 2^2: 2 and 3.
a(3) = 5 = number of squarefree numbers between 3^2 and 4^2: 10, 11, 13, 14 and 15.

Maple

a:= n-> nops(select(numtheory[issqrfree], [$n^2+1..(n+1)^2-1])):
seq(a(n), n=1..80);  # Alois P. Heinz, Jul 16 2019

Mathematica

Table[Count[Range[n^2 + 1, (n + 1)^2 - 1], _?(SquareFreeQ[#] &)], {n, 1, 80}]
(* Harvey P. Dale, Jan 25 2014 *)

Prog

(PARI) a(n)=s=0; for(i=n^2+1, (n+1)^2, if(issquarefree(i), s=s+1)); return(s); \\ corrected by Hugo Pfoertner, Jul 16 2019

Crossrefs

Cf. A005117, A061398.

Sequence in context: A168065 A077724 A163867 * A225636 A023838 A246795

Adjacent sequences: A077378 A077379 A077380 * A077382 A077383 A077384

Keyword

nonn

Author

Amarnath Murthy, Nov 06 2002

Extensions

More terms from Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 23 2004

Name clarified by Hugo Pfoertner, Jul 16 2019

Status

approved