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%I #18 May 22 2017 12:29:49
%S 21,31,109,125,135,151,161,229,281,291,359,369,385,525,541,551,619,
%T 629,645,671,681,749,759,801,879,941,1009,1019,1035,1149,1165,1175,
%U 1399,1425,1435,1529,1539,1555,1565,1581,1669,1685,1695,1799,1851,1919,1945,1971
%N Numbers n such that (n^2+1)/26 is prime.
%C The corresponding primes (n^2+1)/26 are given in A208292(n).
%C a(n) is the smallest positive representative of the class of
%C nontrivial solutions of the congruence x^2==1 (Modd A208292(n)), if n>=2. The trivial solution is the class with representative x=1, which also includes -1. For Modd n see a comment on A203571. For n=1: a(1) = 21 == 13 (Modd 17), and 13 is the smallest positive solution >1.
%C The unique class of nontrivial solutions of the congruence x^2==1 (Modd p), with p an odd prime, exists for any p of the form 4*k+1, given in A002144. Here a subset of these primes is covered, the ones for k=k(n)=(a(n)^2-25)/(4*26). These values are 4, 9, 114, 150, 175, 219, ...
%H Vincenzo Librandi, <a href="/A208293/b208293.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = sqrt(26*A208292(n)-1) = sqrt(8*A208294(n)+1), n>=1.
%e a(3)=109 because (109^2+1)/26 = 457 is prime.
%e 109 = sqrt(26*457-1) = sqrt(8*1485+1).
%t Select[Range[10000], PrimeQ[(#^2 + 1)/26] &] (* _T. D. Noe_, Feb 28 2012 *)
%o (PARI) is(n)=isprime((n^2+1)/26) \\ _Charles R Greathouse IV_, May 22 2017
%Y Cf. A208292, A207337, A207339, A129307, A027862, A002731.
%K nonn
%O 1,1
%A _Wolfdieter Lang_, Feb 27 2012
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