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

Author

Lekraj Beedassy, Feb 25 2002

Keywords

• nice
• nonn

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

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

Crossrefs

Cf. A002110, A005117, A014574, A056911, A065474, A226993, A272799, A280892, A379971 (characteristic function).

Subsequence of A379965.

Programs

Magma

[n: n in [1..110] | IsSquarefree(n-1) and IsSquarefree(n+1)]; // 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 *)

Python

from itertools import count, islice
from sympy import factorint
def A067874_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda k:max(factorint(k-1).values(),default=1)==1 and max(factorint(k+1).values())==1, count(max(startvalue+(startvalue&1),2),2))
A067874_list = list(islice(A067874_gen(),20)) # Chai Wah Wu, Apr 24 2024

Formula

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

Extensions

Corrected and extended by Max Alekseyev, Apr 26 2009

Further edited by Max Alekseyev, Apr 28 2009