A076338 -id:A076338 - cp's OEIS frontend

A076339 Primes of the form 512*k+1.

7681, 10753, 11777, 12289, 13313, 15361, 17921, 18433, 19457, 23041, 25601, 26113, 32257, 36353, 37889, 39937, 40961, 45569, 50177, 51713, 58369, 59393, 61441, 64513, 65537, 67073, 70657, 76289, 76801, 79873, 80897, 81409, 83969

Offset: 1

Reinhard Zumkeller, Oct 07 2002

Odd primes p such that -1 is a 256th power mod p. - Eric M. Schmidt, Mar 27 2014

			A076338(15) = 512*15+1 = a(1) = 7681 = A000040(974);
A076338(21) = 512*21+1 = a(2) = 10753 = A000040(1311);
a(38) - a(37) = 95233 - 87553 = 7680 = a(1)-1.

M. Kraitchik, Theorie des Nombres, Gauthier-Villars (I. 1922, II. 1929).
M. Kraitchik, Recherches sur la theorie des nombres, Gauthier-Villars (1924).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A076338, A065091, A002144, A007519, A094407, A133870, A142925, A208177, A208178.

a076339 n = a076339_list !! (n-1)
a076339_list = filter ((== 1) . a010051) [1,513..]
-- Reinhard Zumkeller, Mar 06 2012

Select[512 Range[164] + 1, PrimeQ] (* Bruno Berselli, Feb 23 2012 *)

forprimestep(p=7681,83969,512, print1(p", ")) \\ Charles R Greathouse IV, Nov 01 2022

a(n) ~ 256n log n. - Charles R Greathouse IV, Nov 01 2022

A158231 a(n) = 256*n + 1.

Original entry on oeis.org

257, 513, 769, 1025, 1281, 1537, 1793, 2049, 2305, 2561, 2817, 3073, 3329, 3585, 3841, 4097, 4353, 4609, 4865, 5121, 5377, 5633, 5889, 6145, 6401, 6657, 6913, 7169, 7425, 7681, 7937, 8193, 8449, 8705, 8961, 9217, 9473, 9729, 9985, 10241, 10497

Offset: 1

Vincenzo Librandi, Mar 14 2009

The identity (256*n + 1)^2 - (256*n^2 + 2*n)*16^2 = 1 can be written as a(n)^2 - A158230(n)*16^2 = 1.

Also the identity (512*n + 1)^2 - (256*n^2 + n)*32^2 = 1 can be written as A076338(n)^2 - (n*a(n))*32^2 = 1. - Vincenzo Librandi, Feb 23 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1
E. J. Barbeau, Polynomial Excursions, Chapter 10: Diophantine equations (2010), pages 84-85 (first identity in the comment section: row 15 in the initial table at p. 85, case d(t) = t*(16^2*t+2)).
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A076338, A158230.

I:=[257, 513]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]];

A158231:=n->256*n + 1; seq(A158231(n), n=1..50); # Wesley Ivan Hurt, Jan 24 2014

256Range[50]+1 (* or *) LinearRecurrence[{2,-1},{257,513},50] (* Harvey P. Dale, Nov 21 2011 *)

PARI
```
a(n) = 256*n + 1
```

a(n) = 2*a(n-1) - a(n-2); a(1)=257, a(2)=513. - Harvey P. Dale, Nov 21 2011

G.f.: x*(257-x)/(x-1)^2. - Harvey P. Dale, Nov 21 2011

A031710 Numbers k such that the least term in the periodic part of the continued fraction for sqrt(k) is 32.

Original entry on oeis.org

257, 1026, 2307, 4100, 6405, 9222, 12551, 16392, 20745, 25610, 30987, 36876, 43277, 50190, 57615, 65552, 74001, 82962, 92435, 102420, 112917, 123926, 135447, 147480, 160025, 173082, 186651, 200732, 215325, 230430, 246047, 262176, 278817, 295970

Offset: 1

David W. Wilson

nonn

The continued fraction expansion of sqrt((k*m)^2+t*m) for m >= 1 where t divides 2*k has the form [k*m, 2*k/t, 2*k*m, 2*k/t, 2*k*m, ...]. Thus numbers of the form (16*m)^2 + m for m >= 1 are in the sequence. Are there any others? - Chai Wah Wu, Jun 18 2016

The term 297058 is not of the form (16*m)^2 + m. - Chai Wah Wu, Jun 19 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 209 terms from Vincenzo Librandi)

Cf. A076338.

Select[Range[10^4], !IntegerQ[Sqrt[#]] && Min[ContinuedFraction[Sqrt[#]][[2]]] == 32 &] (* Vincenzo Librandi, Jun 20 2016 *)

Edited by Charles R Greathouse IV, Aug 09 2010
Incorrect formula and comment removed by Vincenzo Librandi, Jan 09 2012
a(34) from Charles R Greathouse IV, Aug 02 2017

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A076339 Primes of the form 512*k+1.

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A158231 a(n) = 256*n + 1.

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A031710 Numbers k such that the least term in the periodic part of the continued fraction for sqrt(k) is 32.

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