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
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved