A124808 - OEIS

%I #29 Feb 23 2021 05:22:12

%S 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,

%T 25,26,27,28,29,30,30,31,32,33,34,35,35,36,37,37,38,38,39,40,41,42,43,

%U 44,45,46,47,48,49,50,51,51,52,53,54,55,56,57,58,59,60,61,61,62,62,63,64

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

%H Harvey P. Dale, <a href="/A124808/b124808.txt">Table of n, a(n) for n = 0..1000</a>

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

%F 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

%F 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

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

%t Accumulate[Table[If[SquareFreeQ[k^2+1],1,0],{k,0,80}]] (* _Harvey P. Dale_, Mar 04 2014 *)

%t Table[Sum[MoebiusMu[k^2 + 1]^2, {k, 0, n}], {n, 0, 100}] (* _Wesley Ivan Hurt_, Jul 15 2015 *)

%o (PARI) a(n)={my(k,r=0);for(k=0,n,if(issquarefree(k^2+1),r++));return(r);}

%o main(size)=my(n);vector(size,n,a(n-1)) /* _Anders Hellström_, Jul 15 2015 */

%Y Cf. A008683, A049532, A059592, A335963.

%K nonn,easy

%O 0,2

%A _Reinhard Zumkeller_, Nov 08 2006