%I #27 Feb 16 2021 04:32:02

%S 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,

%T 32,33,36,34,37,40,36,43,42,44,46,47,46,49,48,48,51,50,56,55,57,58,60,

%U 63,59,63,63,63,69,70,67,71,71,73,71,74,78,76,78,81,79,84,83,87,85,84,87

%N Number of squarefree numbers between successive squares (exclusive).

%H Hugo Pfoertner, <a href="/A077381/b077381.txt">Table of n, a(n) for n = 1..10000</a>

%H Gabriel Mincu and Laurenţiu Panaitopol, <a href="https://www.jstor.org/stable/43679010">On some properties of squarefree and squareful numbers</a>, Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie, Nouvelle Série, Vol. 49 (97), No. 1 (2006), pp. 63-68; <a href="https://ssmr.ro/bulletin/volumes/49-1/node10.html">alternative link</a>.

%F From _Amiram Eldar_, Feb 16 2021: (Start)

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

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

%e a(1) = 2 because there are 2 squarefree integers between 1^2 and 2^2: 2 and 3.

%e a(3) = 5 = number of squarefree numbers between 3^2 and 4^2: 10, 11, 13, 14 and 15.

%p a:= n-> nops(select(numtheory[issqrfree], [$n^2+1..(n+1)^2-1])):

%p seq(a(n), n=1..80); # _Alois P. Heinz_, Jul 16 2019

%t Table[Count[Range[n^2 + 1, (n + 1)^2 - 1], _?(SquareFreeQ[#] &)], {n, 1, 80}]

%t (* _Harvey P. Dale_, Jan 25 2014 *)

%o (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

%Y Cf. A005117, A061398.

%K nonn

%O 1,1

%A _Amarnath Murthy_, Nov 06 2002

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

%E Name clarified by _Hugo Pfoertner_, Jul 16 2019