A067874 Positive integers x satisfying x^2 - D*y^2 = 1 for a unique integer D.

2, 4, 6, 12, 14, 16, 18, 20, 22, 30, 32, 34, 36, 38, 40, 42, 52, 54, 56, 58, 60, 66, 68, 70, 72, 78, 84, 86, 88, 90, 92, 94, 96, 102, 104, 106, 108, 110, 112, 114, 128, 130, 132, 138, 140, 142, 144, 150, 156, 158, 160, 162, 164, 166, 178, 180, 182, 184, 186, 192, 194, 196, 198

Offset

1,1

Comments

D is unique iff x^2 - 1 is squarefree, in which case it follows with necessity that D=x^2-1 and y=1.

All terms are even. A014574 is a subsequence.

Conjecture: All terms of A002110 > 1 are a subsequence. - Griffin N. Macris, Apr 11 2016

All n such that n+1 and n-1 are in A056911. - Robert Israel, Apr 12 2016

Numbers with a squarefree neighbor. - Juri-Stepan Gerasimov, Jan 17 2017

The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - Amiram Eldar, Feb 25 2021

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Formula

a(n) = 2*A272799(n). - Juri-Stepan Gerasimov, Jan 17 2017

Maple

select(t -> numtheory:-issqrfree(t^2-1), [seq(n, n=2..1000, 2)]); # Robert Israel, Apr 12 2016

Mathematica

Select[Range[200], SquareFreeQ[#^2-1]&] (* Vladimir Joseph Stephan Orlovsky, Oct 26 2009 *)

Prog

(Magma) [n: n in [1..110] | IsSquarefree(n-1) and IsSquarefree(n+1)]; \\ Juri-Stepan Gerasimov, Jan 17 2017

Crossrefs

Cf. A002110, A005117, A014574, A056911, A065474, A226993, A272799, A280892.

Sequence in context: A062856 A056371 A271822 * A015733 A247460 A023187

Adjacent sequences: A067871 A067872 A067873 * A067875 A067876 A067877

Keyword

nice,nonn

Author

Lekraj Beedassy, Feb 25 2002

Extensions

Corrected and extended by Max Alekseyev, Apr 26 2009

Further edited by Max Alekseyev, Apr 28 2009

Status

approved