A069987 Squarefree numbers of form k^2 + 1.

2, 5, 10, 17, 26, 37, 65, 82, 101, 122, 145, 170, 197, 226, 257, 290, 362, 401, 442, 485, 530, 577, 626, 677, 730, 785, 842, 901, 962, 1090, 1157, 1226, 1297, 1370, 1522, 1601, 1765, 1937, 2026, 2117, 2210, 2305, 2402, 2501, 2602, 2705, 2810, 2917, 3026

Offset: 1

Author

Sharon Sela (sharonsela(AT)hotmail.com), May 01 2002

Keywords

• nonn

Comments

Heath-Brown (following Estermann) shows that, for any e > 0, there are k sqrt(x) + O(x^{7/24 + e}) members of this sequence up to x, for k = Product(1 - 2/p^2) = 0.8948412245... (A335963) where the product is over primes p = 1 mod 4. - Charles R Greathouse IV, Nov 19 2012, corrected by Amiram Eldar, Jul 08 2020

Integers k for which the period of the continued fraction of sqrt(k) is 1. - Michel Marcus, Apr 12 2019

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• T. Estermann, Einige Sätze über quadratfreie Zahlen, Math. Ann. 105 (1931), pp. 653-662.
• D. R. Heath-Brown, Square-free values of n^2 + 1, arXiv:1010.6217 [math.NT], 2010-2012.
• D. R. Heath-Brown, Square-free values of n^2 + 1, Acta Arithmetica 155 (2012), pp. 1-13.

Crossrefs

Cf. A059591, A002496, A124809, A005117, A002522, A335963.

Programs

Maple

select(numtheory:-issqrfree, [seq(n^2+1, n=1..100)]); # Robert Israel, Feb 09 2016

Mathematica

Select[ Range[10^4], IntegerQ[ Sqrt[ # - 1]] && Union[ Transpose[ FactorInteger[ # ]] [[2]]] [[ -1]] == 1 &]
Select[Range[60]^2+1,SquareFreeQ] (* Harvey P. Dale, Mar 21 2013 *)

PARI

for(n=1,100,if(issquarefree(n^2+1),print1(n^2+1,",")))

Formula

a(n) = A049533(n)^2 + 1.

Extensions

Edited and extended by Robert G. Wilson v, Benoit Cloitre and Vladeta Jovovic, May 04 2002