cp's OEIS Frontend

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

%I A208292 #18 Aug 03 2014 14:01:39
%S A208292 17,37,457,601,701,877,997,2017,3037,3257,4957,5237,5701,10601,11257,
%T A208292 11677,14737,15217,16001,17317,17837,21577,22157,24677,29717,34057,
%U A208292 39157,39937,41201,50777,52201,53101,75277,78101,79201,89917,91097,93001,94201,96137
%N A208292 Primes of the form (n^2+1)/26.
%C A208292 Equivalently, primes of the form (K^2 + (K+1)^2)/13. The
%C A208292 connection to the primes of the form (m^2+1)/26 is given by m=2*K+1 (m is necessarily odd).
%C A208292 The corresponding m=m(n) values are given in A208293(n).
%C A208292 Equivalently, primes of the form (4*T(K)+1)/13, with the
%C A208292   corresponding triangular numbers T(K):=A000217(K), for
%C A208292   K=K(n)=(m(n)-1)/2, given in A208294(n).
%C A208292 For n>=2 the smallest positive representative of the class of
%C A208292   nontrivial solutions of the congruence x^2==1 (Modd a(n)) is
%C A208292   x=m(n). The trivial solution is the class with representative x=1, which also includes -1. For the prime
%C A208292   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.
%C A208292 These primes with corresponding m values are such that floor(m(n)^2/p(n)) = 5^2, n>=1.
%H A208292 Vincenzo Librandi, <a href="/A208292/b208292.txt">Table of n, a(n) for n = 1..2000</a>
%F A208292 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).
%e A208292 a(3)=457, m(3)=A208293(3)=109. T(K(3))=A000217((109-1)/2)=
%e A208292   A000217(54)=A208294(3)=1485.
%t A208292 Select[(Range[2000]^2 + 1)/26, PrimeQ] (* _T. D. Noe_, Feb 28 2012 *)
%Y A208292 Cf. A207337, A207339 (case floor(m^2/p)=3^2); A129307, A027862, A002731 (case floor(m^2/p)=1^2).
%K A208292 nonn
%O A208292 1,1
%A A208292 _Wolfdieter Lang_, Feb 27 2012