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

1,1

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.

Formula

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A322008 A300164 A248193 * A248742 A246884 A119114

Adjacent sequences: A069984 A069985 A069986 * A069988 A069989 A069990

Keyword

nonn

Author

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

Extensions

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

Status

approved