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

%I A067874 #59 Jan 24 2025 13:26:19
%S A067874 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 A067874 72,78,84,86,88,90,92,94,96,102,104,106,108,110,112,114,128,130,132,
%U A067874 138,140,142,144,150,156,158,160,162,164,166,178,180,182,184,186,192,194,196,198
%N A067874 Positive integers x satisfying x^2 - D*y^2 = 1 for a unique integer D.
%C A067874 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 A067874 All terms are even. A014574 is a subsequence.
%C A067874 Conjecture: All terms of A002110 > 1 are a subsequence. - _Griffin N. Macris_, Apr 11 2016
%C A067874 All n such that n+1 and n-1 are in A056911. - _Robert Israel_, Apr 12 2016
%C A067874 The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - _Amiram Eldar_, Feb 25 2021
%H A067874 T. D. Noe, <a href="/A067874/b067874.txt">Table of n, a(n) for n = 1..1000</a>
%F A067874 a(n) = 2*A272799(n). - _Juri-Stepan Gerasimov_, Jan 17 2017
%p A067874 select(t -> numtheory:-issqrfree(t^2-1), [seq(n,n=2..1000,2)]); # _Robert Israel_, Apr 12 2016
%t A067874 Select[Range[200], SquareFreeQ[#^2-1]&] (* _Vladimir Joseph Stephan Orlovsky_, Oct 26 2009 *)
%o A067874 (Magma) [n: n in [1..110] | IsSquarefree(n-1) and IsSquarefree(n+1)]; // _Juri-Stepan Gerasimov_, Jan 17 2017
%o A067874 (Python)
%o A067874 from itertools import count, islice
%o A067874 from sympy import factorint
%o A067874 def A067874_gen(startvalue=2): # generator of terms >= startvalue
%o A067874     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 A067874_list = list(islice(A067874_gen(),20)) # _Chai Wah Wu_, Apr 24 2024
%Y A067874 Cf. A002110, A005117, A014574, A056911, A065474, A226993, A272799, A280892, A379971 (characteristic function).
%Y A067874 Subsequence of A379965.
%K A067874 nice,nonn
%O A067874 1,1
%A A067874 _Lekraj Beedassy_, Feb 25 2002
%E A067874 Corrected and extended by _Max Alekseyev_, Apr 26 2009
%E A067874 Further edited by _Max Alekseyev_, Apr 28 2009