login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A255247 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A001132(n), n>=1 (primes congruent to {1,7} mod 8). 5
5, 9, 7, 13, 11, 9, 21, 13, 11, 19, 25, 17, 15, 29, 21, 19, 15, 31, 23, 37, 17, 35, 27, 41, 25, 33, 23, 21, 29, 37, 49, 23, 21, 41, 47, 39, 29, 37, 25, 23, 57, 35, 43, 33, 49, 55, 27, 59, 65, 33, 51, 43, 31, 29, 41, 49, 69, 55, 53, 29, 43, 59, 51, 41, 37, 35 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
For the corresponding term y2(n) see A255248(n).
For the positive fundamental proper (sometimes called primitive) solutions x1(n) and y1(n) of the first class of this (generalized) Pell equation see A255235(n) and A255246(n).
The present solutions of this second class are the next to smallest positive ones. Note that for prime 2 only the first class exists.
For the derivation based on the book of Nagell see the comments on A254934 and A254938 for the primes 1 (mod 8) and 7 (mod 8) separately, where also the Nagell reference is given.
LINKS
FORMULA
a(n)^2 - 2*A255248(n)^2 = -A001132(n), n >= 1, gives the second smallest positive (proper) solution of this (generalized) Pell equation.
a(n) = -(3*A255235(n+1) - 4*A255246(n+1)), n >= 1.
EXAMPLE
The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are
the prime A001132(n) is listed as first entry):
[7, [5, 4]], [17, [9, 7]], [23, [7, 6]],
[31, [13, 10]], [41, [11, 9]], [47, [9, 8]],
[71, [21, 16]], [73, [13, 11]], [79, [11, 10],
[89, [19, 15]], [97, [25, 19]], [103, [17, 14]],
[113, [15, 13]], [127, [29, 22]],
[137, [21, 17]], [151, [19, 16]],
[167, [15, 14]], [191, [31, 24]],
[193, [23, 19]], [199, [37, 28]],
[223, [17, 16]], [233, [35, 27]],
[239, [27, 22]], [241, [41, 31]], ...
n = 1: 5^2 - 2*4^2 = 25 - 32 = -7 = -A001132(1).
a(3) = -(3*3 - 4*4) = 16 - 9 = 7.
CROSSREFS
Sequence in context: A232190 A160050 A055566 * A366841 A153610 A357464
KEYWORD
nonn
AUTHOR
Wolfdieter Lang, Feb 19 2015
EXTENSIONS
More terms from Colin Barker, Feb 26 2015
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 01:22 EDT 2024. Contains 370952 sequences. (Running on oeis4.)