A092575 -id:A092575 - cp's OEIS frontend

A092572 Numbers of the form x^2 + 3y^2 where x and y are positive integers.

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

Eric W. Weisstein, Feb 28 2004

nonn

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

			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.

Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem. New York: Springer-Verlag (1979): 4.

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

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

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

Union[Flatten[Table[a^2 + 3b^2, {a, 20}, {b, Ceiling[Sqrt[(400 - a^2)/3]]}]]] (* Alonso del Arte, Jul 19 2012 *)

A092573 Number of solutions to x^2 + 3y^2 = n in positive integers x and y.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 3, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0

Offset: 0

Eric W. Weisstein, Feb 28 2004

nonn

Robert Israel, Table of n, a(n) for n = 0..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, Euler's 6n+1 Theorem

Cf. A002476, A092572, A092575.

N:= 300: # to get a(0)..a(N)
V:= Vector(N):
for y from 1 to floor(sqrt(N/3-1)) do
  js:= [seq(x^2+3*y^2, x=1..floor(sqrt(N-3*y^2)))];
  V[js]:= map(`+`,V[js],1);
od:
0,op(convert(V,list)); # Robert Israel, Apr 03 2017

r[z_] := Reduce[x > 0 && y > 0 && x^2 + 3 y^2 == z, {x, y}, Integers]; Table[rz = r[z]; If[rz === False, 0, If[rz[[0]] === Or, Length[rz], 1]], {z, 0, 102}] (* Jean-François Alcover, Oct 23 2012 *)
gf = (EllipticTheta[3, 0, x]-1)*(EllipticTheta[3, 0, x^3]-1)/4 + O[x]^105;
CoefficientList[gf, x] (* Jean-François Alcover, Jul 02 2018, after Robert Israel *)

a(n) = ( A033716(n) - A000122(n) - A000122(n/3) + A000007(n) )/4. - Max Alekseyev, Sep 29 2012

G.f.: (Theta_3(0,x)-1)*(Theta_3(0,x^3)-1)/4 where Theta_3 is a Jacobi theta function. - Robert Israel, Apr 03 2017

Definition corrected by David A. Corneth, Apr 03 2017

A092574 Positive integers that can be represented in the form x^2 + 3y^2 with (x,y) = 1 and x and y positive.

Original entry on oeis.org

4, 7, 12, 13, 19, 21, 28, 31, 37, 39, 43, 49, 52, 57, 61, 67, 73, 76, 79, 84, 91, 93, 97, 103, 109, 111, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 172, 181, 183, 193, 196, 199, 201, 211, 217, 219, 223, 228, 229, 237, 241, 244, 247, 259, 268

Offset: 1

Eric W. Weisstein, Feb 28 2004

nonn

Superset of primes of the form 6n+1 (A002476).

For all proper solutions with nonnegative x and y see A244819. - Wolfdieter Lang, Mar 02 2021

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

Cf. A002476, A092572, A092573, A092575, A244819.

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A092572 Numbers of the form x^2 + 3y^2 where x and y are positive integers.

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A092573 Number of solutions to x^2 + 3y^2 = n in positive integers x and y.

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A092574 Positive integers that can be represented in the form x^2 + 3y^2 with (x,y) = 1 and x and y positive.

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