The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A069987 Squarefree numbers of form k^2 + 1. 10
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A322008 A300164 A248193 * A248742 A246884 A119114
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 19 14:45 EDT 2024. Contains 372698 sequences. (Running on oeis4.)