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A002733 Numbers k such that (k^2 + 1)/10 is prime.

%I A002733 M4342 N1047 #36 Feb 28 2022 12:09:31
%S A002733 7,13,17,23,27,33,37,53,63,67,77,87,97,103,113,127,137,147,153,163,
%T A002733 167,197,223,227,247,263,267,277,283,287,297,303,323,347,363,367,373,
%U A002733 383,397
%N A002733 Numbers k such that (k^2 + 1)/10 is prime.
%C A002733 Contribution from _Wolfdieter Lang_, Feb 27 2012: (Start)
%C A002733 The corresponding primes (n^2 + 1)/10 are given in A207337(n).
%C A002733 a(n) is the smallest positive representative of the class of nontrivial solutions of the congruence x^2 == 1 (Modd A207337(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) = 7 == 3 (Modd 5), and 3 is the smallest positive solution > 1.
%C A002733 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 - 9)/40. These k-values are [1, 4, 7, 13, 18, 27, 34, 70, 99, 112, ...].
%C A002733 (End)
%D A002733 L. Euler, De numeris primis valde magnis (E283), reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 3, p. 25.
%D A002733 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002733 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002733 Reinhard Zumkeller, <a href="/A002733/b002733.txt">Table of n, a(n) for n = 1..10000</a>
%H A002733 L. Euler, <a href="http://eulerarchive.maa.org/pages/E283.html">De numeris primis valde magnis (E283)</a>, The Euler Archive
%F A002733 a(n) = sqrt(10*A207337(n)-1) = sqrt(8*A207339(n)+1), n >= 1. - _Wolfdieter Lang_, Feb 27 2012
%p A002733 a := [ ]: for n from 1 to 400 do if (n^2+1 mod 10) = 0 and isprime((n^2+1)/10) then a := [ op(a), n ]; fi; od;
%t A002733 Select[Range[573], PrimeQ[(#^2 + 1)/10] &] (* _T. D. Noe_, Feb 28 2012 *)
%o A002733 (PARI) forstep(n=7,1e3,[6,4],if(isprime(n^2\10+1),print1(n", "))) \\ _Charles R Greathouse IV_, Mar 11 2012
%o A002733 (Haskell)
%o A002733 a002733 = a000196 . (subtract 1) . (* 10) . a207337
%o A002733 -- _Reinhard Zumkeller_, Apr 06 2012
%Y A002733 Cf. A000196, A010051, A002522, A002731, A002732.
%K A002733 nonn
%O A002733 1,1
%A A002733 _N. J. A. Sloane_