A001912 Numbers k such that 4*k^2 + 1 is prime. (Formerly M0636 N0232)

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, 73, 75, 78, 80, 85, 88, 90, 92, 102, 103, 105, 112, 115, 118, 120, 125, 128, 130, 132, 135, 140, 142, 150, 153, 157, 163, 170, 175, 192, 193, 198, 200

Offset

1,2

Comments

Complement of A094550. - Hermann Stamm-Wilbrandt, Sep 16 2014

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

4*a(n)^2 + 1 = A002496(n+1). - Hal M. Switkay, Apr 03 2022

References

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

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

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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]

E. Kogbetliantz and A. Krikorian Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971 [Annotated scans of a few pages]

Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Formula

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

Maple

A001912 := proc(n)
    option remember;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isprime(4*a^2+1) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Aug 09 2012

Mathematica

Select[Range[200], PrimeQ[4#^2 + 1] &] (* Alonso del Arte, Dec 20 2013 *)

Prog

(Magma) [n: n in [1..100] | IsPrime(4*n^2+1)] // Vincenzo Librandi, Nov 21 2010
(PARI) is(n)=isprime(4*n^2 + 1) \\ Charles R Greathouse IV, Apr 28 2015

Crossrefs

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

Sequence in context: A186317 A285077 A143826 * A186221 A083027 A060107

Adjacent sequences: A001909 A001910 A001911 * A001913 A001914 A001915

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved