A124808 Number of numbers k <= n such that k^2 + 1 is squarefree.

1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 35, 36, 37, 37, 38, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 61, 62, 62, 63, 64

Offset: 0

Author

Reinhard Zumkeller, Nov 08 2006

Keywords

• nonn
• easy

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A008683, A049532, A059592, A335963.

Programs

Maple

ListTools:-PartialSums([seq(numtheory:-mobius(k^2+1)^2, k=0..100)]); # Robert Israel, Jul 15 2015

Mathematica

Accumulate[Table[If[SquareFreeQ[k^2+1],1,0],{k,0,80}]] (* Harvey P. Dale, Mar 04 2014 *)
Table[Sum[MoebiusMu[k^2 + 1]^2, {k, 0, n}], {n, 0, 100}] (* Wesley Ivan Hurt, Jul 15 2015 *)

PARI

a(n)={my(k,r=0);for(k=0,n,if(issquarefree(k^2+1),r++));return(r);}
main(size)=my(n);vector(size,n,a(n-1)) /* Anders Hellström, Jul 15 2015 */

Formula

a(0) = 1, a(n) = a(n-1) + 0^(A059592(n) - 1).

a(n) = Sum_{k=0..n} mu(k^2+1)^2, where mu(n) is the Mobius function (A008683). - Wesley Ivan Hurt, Jul 15 2015

a(n) ~ c*n where c = Product_{p prime, p == 1 (mod 4)} (1 - 2/p^2) = 0.894841... (A335963). - Amiram Eldar, Feb 23 2021