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

Author

Amarnath Murthy, Nov 06 2002

Keywords

• nonn

Examples

			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.
		

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.

Crossrefs

Cf. A005117, A061398.

Programs

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 *)

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

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)

Extensions

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

Name clarified by Hugo Pfoertner, Jul 16 2019