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%I M2444 N0971 #54 Oct 28 2023 13:08:43
%S 3,5,9,11,15,19,25,29,35,39,45,49,51,59,61,65,69,71,79,85,95,101,121,
%T 131,139,141,145,159,165,169,171,175,181,195,199,201,205,209,219,221,
%U 231,245,261,271,275,279,289,299,309,315,321,325,329,335,345,349,371,375,379,391,399,405
%N Numbers k such that (k^2 + 1)/2 is prime.
%C From _Wolfdieter Lang_, Feb 24 2012: (Start)
%C a(n) = sqrt(8*A129307(n)+1) = sqrt(2*A027862(n)-1), n >= 1.
%C a(n) is the nontrivial solution of the congruence a(n)^2 == 1 (Modd A027862(n)). The trivial one is +1. For Modd n see a comment on A203571. E.g., a(3)^2 = 81 == 1 (Modd 41), see a comment on A027862.
%C (End)
%D L. Euler, De numeris primis valde magnis (E283), reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 3, p. 24.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Ray Chandler, <a href="/A002731/b002731.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H L. Euler, <a href="http://eulerarchive.maa.org/pages/E283.html">De numeris primis valde magnis (E283)</a>, The Euler Archive.
%t Select[Range[400], PrimeQ[(#^2 + 1)/2] &] (* _Alonso del Arte_, Feb 24 2012 *)
%o (PARI)
%o forstep(n=1,10^3,2, if(isprime((n^2+1)/2),print1(n,", ")));
%o /* _Joerg Arndt_, Sep 02 2012 */
%o (Magma) [n: n in [3..410] | IsPrime((n^2+1) div 2) ]; // _Vincenzo Librandi_, Sep 25 2012
%o (Haskell)
%o a002731 n = a002731_list !! (n-1)
%o a002731_list = filter ((== 1) . a010051 . a000982) [1, 3 ..]
%o -- _Reinhard Zumkeller_, Jul 13 2014
%Y Equals 2*A027861(n-1)+1. A027862 gives primes, A091277 gives prime index.
%Y Cf. A000982, A010051, A116945, A129307, A188546, A188547, A187431.
%K nonn,easy,nice
%O 1,1
%A _N. J. A. Sloane_
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