OFFSET
1,1
COMMENTS
Superset of primes of the form 6n+1 (A002476).
It seems that all integer solutions of ((a+b)^3 - (a-b)^3) / (2*b) = c^3 have c = x^2 + 3*y^2. - Juergen Buchmueller (pullmoll(AT)t-online.de), Apr 04 2008
To prove the case of cubes in Fermat's last theorem, Euler considered numbers of the form a^2 + 3b^2. In the equation x^3 + y^3 = z^3, Euler specified that x = a - b and y = a + b. - Alonso del Arte, Jul 19 2012
All terms == 0,1,3,4, or 7 (mod 9). - Robert Israel, Apr 03 2017
REFERENCES
Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem. New York: Springer-Verlag (1979): 4.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
E. Akhtarkavan, M. F. M. Salleh and O. Sidek, Multiple Descriptions Video Coding Using Coinciding Lattice Vector Quantizer for H.264/AVC and Motion JPEG2000, World Applied Sciences Journal 21 (2): 157-169, 2013. - From N. J. A. Sloane, Feb 11 2013
Eric Weisstein's World of Mathematics, Eulers 6n Plus 1 Theorem
EXAMPLE
7 is of the specified form, since 2^2 + 3 * 1^2 = 7.
So is 12, since 3^2 + 3 * 1^2 = 12, and 13, with 1^2 + 3 * 2^2 = 13.
MAPLE
N:= 1000: # to get all terms <= N
S:= {seq(seq(x^2 + 3*y^2, x = 1 .. floor(sqrt(N - 3*y^2))),
y=1..floor(sqrt(N/3-1)))}:
sort(convert(S, list)); # Robert Israel, Apr 03 2017
MATHEMATICA
Union[Flatten[Table[a^2 + 3b^2, {a, 20}, {b, Ceiling[Sqrt[(400 - a^2)/3]]}]]] (* Alonso del Arte, Jul 19 2012 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Feb 28 2004
STATUS
approved