%I M0636 N0232 #64 Apr 05 2022 21:15:10

%S 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 73,75,78,80,85,88,90,92,102,103,105,112,115,118,120,125,128,130,132,

%U 135,140,142,150,153,157,163,170,175,192,193,198,200

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

%C Complement of A094550. - _Hermann Stamm-Wilbrandt_, Sep 16 2014

%C 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 4*a(n)^2 + 1 = A002496(n+1). - _Hal M. Switkay_, Apr 03 2022

%D E. Kogbetliantz and A. Krikorian, Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971, p. 1.

%D M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 11.

%D C. S. Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 116.

%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 T. D. Noe, <a href="/A001912/b001912.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 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 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 a(n) = A005574(n+1)/2.

%p A001912 := proc(n)

%p option remember;

%p if n = 1 then

%p 1;

%p else

%p for a from procname(n-1)+1 do

%p if isprime(4*a^2+1) then

%p return a;

%p end if;

%p end do:

%p end if;

%p end proc: # _R. J. Mathar_, Aug 09 2012

%t Select[Range[200], PrimeQ[4#^2 + 1] &] (* _Alonso del Arte_, Dec 20 2013 *)

%o (Magma) [n: n in [1..100] | IsPrime(4*n^2+1)] // _Vincenzo Librandi_, Nov 21 2010

%o (PARI) is(n)=isprime(4*n^2 + 1) \\ _Charles R Greathouse IV_, Apr 28 2015

%Y Cf. A002496, A005574, A062325, A090693, A094550, A214517 (first differences).

%K nonn,easy,nice

%O 1,2

%A _N. J. A. Sloane_