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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 (list; graph; refs; listen; history; text; internal format)
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

Cf. A002476, A092573, A092575, A158937 (similar definition but with duplicates left in).

Sequence in context: A249918 A340244 A158937 * A092574 A310769 A058271

Adjacent sequences:  A092569 A092570 A092571 * A092573 A092574 A092575

KEYWORD

nonn

AUTHOR

Eric W. Weisstein, Feb 28 2004

STATUS

approved

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Last modified May 9 19:07 EDT 2021. Contains 343746 sequences. (Running on oeis4.)