A208292 Primes of the form (n^2+1)/26.

17, 37, 457, 601, 701, 877, 997, 2017, 3037, 3257, 4957, 5237, 5701, 10601, 11257, 11677, 14737, 15217, 16001, 17317, 17837, 21577, 22157, 24677, 29717, 34057, 39157, 39937, 41201, 50777, 52201, 53101, 75277, 78101, 79201, 89917, 91097, 93001, 94201, 96137

Offset

1,1

Comments

Equivalently, primes of the form (K^2 + (K+1)^2)/13. The

connection to the primes of the form (m^2+1)/26 is given by m=2*K+1 (m is necessarily odd).

The corresponding m=m(n) values are given in A208293(n).

Equivalently, primes of the form (4*T(K)+1)/13, with the

corresponding triangular numbers T(K):=A000217(K), for

K=K(n)=(m(n)-1)/2, given in A208294(n).

For n>=2 the smallest positive representative of the class of

nontrivial solutions of the congruence x^2==1 (Modd a(n)) is

x=m(n). The trivial solution is the class with representative x=1, which also includes -1. For the prime

a(1)=17 the nontrivial solution is 13 (see A002733(2)). Unique nontrivial smallest positive representatives exist for the solutions for any prime of the form 4*k+1, given in A002144. Here the subset with k=k(n)=(a(n)-1)/4 appears, namely 4,9,114,150,175,219,.... For Modd n see a comment on A203571.

These primes with corresponding m values are such that floor(m(n)^2/p(n)) = 5^2, n>=1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Formula

a(n) is the n-th member of the increasingly ordered list of primes of the form (m^2+1)/10, where m=m(n) is necessarily an odd integer, the positive one is A208293(n).

Example

a(3)=457, m(3)=A208293(3)=109. T(K(3))=A000217((109-1)/2)=
  A000217(54)=A208294(3)=1485.

Mathematica

Select[(Range[2000]^2 + 1)/26, PrimeQ] (* T. D. Noe, Feb 28 2012 *)

Crossrefs

Cf. A207337, A207339 (case floor(m^2/p)=3^2); A129307, A027862, A002731 (case floor(m^2/p)=1^2).

Sequence in context: A157467 A093343 A153685 * A121710 A051779 A139579

Adjacent sequences: A208289 A208290 A208291 * A208293 A208294 A208295

Keyword

nonn

Author

Wolfdieter Lang, Feb 27 2012

Status

approved