A067874 - OEIS

%I #56 Jun 03 2024 12:16:27

%S 2,4,6,12,14,16,18,20,22,30,32,34,36,38,40,42,52,54,56,58,60,66,68,70,

%T 72,78,84,86,88,90,92,94,96,102,104,106,108,110,112,114,128,130,132,

%U 138,140,142,144,150,156,158,160,162,164,166,178,180,182,184,186,192,194,196,198

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

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

%C All terms are even. A014574 is a subsequence.

%C Conjecture: All terms of A002110 > 1 are a subsequence. - _Griffin N. Macris_, Apr 11 2016

%C All n such that n+1 and n-1 are in A056911. - _Robert Israel_, Apr 12 2016

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

%H T. D. Noe, <a href="/A067874/b067874.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = 2*A272799(n). - _Juri-Stepan Gerasimov_, Jan 17 2017

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

%t Select[Range[200], SquareFreeQ[#^2-1]&] (* _Vladimir Joseph Stephan Orlovsky_, Oct 26 2009 *)

%o (Magma) [n: n in [1..110] | IsSquarefree(n-1) and IsSquarefree(n+1)]; // _Juri-Stepan Gerasimov_, Jan 17 2017

%o (Python)

%o from itertools import count, islice

%o from sympy import factorint

%o def A067874_gen(startvalue=2): # generator of terms >= startvalue

%o     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))

%o A067874_list = list(islice(A067874_gen(),20)) # _Chai Wah Wu_, Apr 24 2024

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

%K nice,nonn

%O 1,1

%A _Lekraj Beedassy_, Feb 25 2002

%E Corrected and extended by _Max Alekseyev_, Apr 26 2009

%E Further edited by _Max Alekseyev_, Apr 28 2009