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A001912 Numbers k such that 4*k^2 + 1 is prime.

%I A001912 M0636 N0232 #64 Apr 05 2022 21:15:10
%S A001912 1,2,3,5,7,8,10,12,13,18,20,27,28,33,37,42,45,47,55,58,60,62,63,65,67,
%T A001912 73,75,78,80,85,88,90,92,102,103,105,112,115,118,120,125,128,130,132,
%U A001912 135,140,142,150,153,157,163,170,175,192,193,198,200
%N A001912 Numbers k such that 4*k^2 + 1 is prime.
%C A001912 Complement of A094550. - _Hermann Stamm-Wilbrandt_, Sep 16 2014
%C A001912 Positive integers whose square is the sum of two triangular numbers in exactly one way (A000217(k) + A000217(k+1) = k*(k+1)/2 + (k+1)*(k+2)/2 = (k+1)^2). In other words, positive integers k such that A052343(k^2) = 1. - _Altug Alkan_, Jul 06 2016
%C A001912 4*a(n)^2 + 1 = A002496(n+1). - _Hal M. Switkay_, Apr 03 2022
%D A001912 E. Kogbetliantz and A. Krikorian, Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971, p. 1.
%D A001912 M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 11.
%D A001912 C. S. Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 116.
%D A001912 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001912 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001912 T. D. Noe, <a href="/A001912/b001912.txt">Table of n, a(n) for n = 1..10000</a>
%H A001912 A. J. C. Cunningham, <a href="/A001912/a001912.pdf">Binomial Factorisations</a>, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
%H A001912 E. Kogbetliantz and A. Krikorian <a href="/A002970/a002970.pdf">Handbook of First Complex Prime Numbers</a>, Gordon and Breach, NY, 1971 [Annotated scans of a few pages]
%H A001912 Marek Wolf, <a href="https://arxiv.org/abs/0803.1456">Search for primes of the form m^2+1</a>, arXiv:0803.1456 [math.NT], 2008-2010.
%F A001912 a(n) = A005574(n+1)/2.
%p A001912 A001912 := proc(n)
%p A001912     option remember;
%p A001912     if n = 1 then
%p A001912         1;
%p A001912     else
%p A001912         for a from procname(n-1)+1 do
%p A001912             if isprime(4*a^2+1) then
%p A001912                 return a;
%p A001912             end if;
%p A001912         end do:
%p A001912     end if;
%p A001912 end proc: # _R. J. Mathar_, Aug 09 2012
%t A001912 Select[Range[200], PrimeQ[4#^2 + 1] &] (* _Alonso del Arte_, Dec 20 2013 *)
%o A001912 (Magma) [n: n in [1..100] | IsPrime(4*n^2+1)] // _Vincenzo Librandi_, Nov 21 2010
%o A001912 (PARI) is(n)=isprime(4*n^2 + 1) \\ _Charles R Greathouse IV_, Apr 28 2015
%Y A001912 Cf. A002496, A005574, A062325, A090693, A094550, A214517 (first differences).
%K A001912 nonn,easy,nice
%O A001912 1,2
%A A001912 _N. J. A. Sloane_