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A067874
Positive integers x satisfying x^2 - D*y^2 = 1 for a unique integer D.
9
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
The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - Amiram Eldar, Feb 25 2021
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
(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
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