A049532 Numbers k such that k^2 + 1 is not squarefree.

7, 18, 32, 38, 41, 43, 57, 68, 70, 82, 93, 99, 107, 117, 118, 132, 143, 157, 168, 182, 193, 207, 218, 232, 239, 243, 251, 257, 268, 282, 293, 307, 318, 327, 332, 343, 357, 368, 378, 382, 393, 407, 408, 418, 432, 437, 443, 457, 468, 482, 493, 500, 507, 515

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

The sequence is infinite. For instance, it contains all numbers of the form 7 + 25m. - Emmanuel Vantieghem, Oct 25 2016

More generally, the sequence contains all numbers of the form a(n) + (a(n)^2 + 1) * m for even a(n) and a(n) + (a(n)^2 + 1) * m / 2 for odd a(n). - David A. Corneth, Oct 25 2016

The asymptotic density of this sequence is 1 - A335963 = 0.1051587754... - Amiram Eldar, Jul 08 2020

Examples

			a(1) = 7 because 7^2 + 1 = 49 + 1 = 50 is divisible by 25, a square.
		

Links

• R. J. Mathar, Table of n, a(n) for n = 1..7999

Crossrefs

Cf. A002522, A059592, A124809, A335963.

Programs

Magma

[n: n in [1..6*10^2]| not IsSquarefree(n^2+1)]; // Bruno Berselli, Oct 15 2012

Mathematica

n=1;Reap[Do[While[SquareFreeQ[n^2+1],n++];Sow[n];n++,{c,10000}]][[2,1]] (* Zak Seidov, Feb 24 2011 *)

PARI

for(n=1,1e4,if(!issquarefree(n^2+1),print1(n", "))) \\ Charles R Greathouse IV, Feb 24 2011

Formula

A059592(a(n)) > 1; A124809(n) = a(n)^2 + 1. - Reinhard Zumkeller, Nov 08 2006

Extensions

Definition rewritten by Bruno Berselli, Oct 15 2012

Mathematica updated by Jean-François Alcover, Jun 19 2013