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

5, 17, 29, 53, 73, 109, 137, 281, 397, 449, 593, 757, 941, 1061, 1277, 1613, 1877, 2161, 2341, 2657, 2789, 3881, 4973, 5153, 6101, 6917, 7129, 7673, 8009, 8237, 8821, 9181, 10433, 12041, 13177, 13469, 13913, 14669, 15761, 17389, 18233, 18749

Offset: 1

Author

Wolfdieter Lang, Feb 27 2012

Keywords

• nonn

Comments

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).

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).

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A207339, A129307, A027862, A002731.

Cf. A010051, A002522.

Programs

Haskell

a207337 n = a207337_list !! (n-1)
a207337_list = f a002522_list where
   f (x:xs) | m == 0 && a010051 y == 1 = y : f xs
            | otherwise                = f xs
            where (y,m) = divMod x 10
-- Reinhard Zumkeller, Apr 06 2012

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, namely A002733(n).