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A092572
Numbers of the form x^2 + 3y^2 where x and y are positive integers.
34
4, 7, 12, 13, 16, 19, 21, 28, 31, 36, 37, 39, 43, 48, 49, 52, 57, 61, 63, 64, 67, 73, 76, 79, 84, 91, 93, 97, 100, 103, 108, 109, 111, 112, 117, 124, 127, 129, 133, 139, 144, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192, 193, 196, 199
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
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
Cf. A002476, A092573, A092575, A158937 (similar definition but with duplicates left in).
Sequence in context: A249918 A340244 A158937 * A344159 A092574 A310769
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Feb 28 2004
STATUS
approved