cp's OEIS Frontend

A207337 Primes of the form (m^2+1)/10.

%I A207337 #21 Jul 13 2013 12:04:22
%S A207337 5,17,29,53,73,109,137,281,397,449,593,757,941,1061,1277,1613,1877,
%T A207337 2161,2341,2657,2789,3881,4973,5153,6101,6917,7129,7673,8009,8237,
%U A207337 8821,9181,10433,12041,13177,13469,13913,14669,15761,17389,18233,18749
%N A207337 Primes of the form (m^2+1)/10.
%C A207337 Equivalently, primes of the form (K^2 + (K+1)^2)/5. The connection to the primes of the form (m^2+1)/10 is given by m=2*K+1 (m is necessarily odd). The corresponsding m=m(n) values are given in A002733(n).
%C A207337 Equivalently, primes of the form (4*T(K)+1)/5, with the corresponding triangular numbers T(K):=A000217(K), for K(n)=(m(n)-1)/2, given in A207339(n).
%C A207337 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)=5 the smallest positive nontrivial solution is 3 (see A027862(1) with  A002731(1)). Such a nontrivial smallest positive representative exists for each unique class of solutions of this congruence Modd p for any prime p of the form 4*k+1, given in A002144. Here the subset with k=k(n)=(a(n)-1)/4 appears, namely 1, 4, 7, 13, 18, 27, 34, 70,... For Modd n see a comment on A203571.
%H A207337 Reinhard Zumkeller, <a href="/A207337/b207337.txt">Table of n, a(n) for n = 1..10000</a>
%F A207337 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, namely A002733(n).
%e A207337 a(3)=29, m(3)=A002733(3)=17. T(K(3))=A000217((17-1)/2)= A000217(8)=A207339(3)=36. (8^2+9^2)/5 = 29 = (4*36+1)/5.
%o A207337 (Haskell)
%o A207337 a207337 n = a207337_list !! (n-1)
%o A207337 a207337_list = f a002522_list where
%o A207337    f (x:xs) | m == 0 && a010051 y == 1 = y : f xs
%o A207337             | otherwise                = f xs
%o A207337             where (y,m) = divMod x 10
%o A207337 -- _Reinhard Zumkeller_, Apr 06 2012
%Y A207337 Cf.  A207339, A129307, A027862, A002731.
%Y A207337 Cf. A010051, A002522.
%K A207337 nonn
%O A207337 1,1
%A A207337 _Wolfdieter Lang_, Feb 27 2012