A281391 Vinogradov's number J_{3,2}(n).

1, 20, 93, 256, 563, 1032, 1771, 2744, 4077, 5788, 7985, 10560, 13855, 17600, 22047, 27304, 33425, 40140, 47989, 56504, 66315, 77296, 89411, 102336, 117061, 132956, 150201, 168904, 189479, 211080, 235111, 260240, 287385, 316420, 347237

Offset: 1

Author

Juan Arias-de-Reyna, Jan 21 2017

Keywords

• nonn

Comments

a(n) is the number of solutions of the system of equations x_1 + x_2 + x_3 = y_1 + y_2 + y_3, x_1^2 + x_2^2 + x_3^2 = y_1^2 + y_2^2 + y_3^2 and such that 1 <= x_1, x_2, x_3, y_1, y_2, y_3 <= n.

Vinogradov's numbers J_{s,k}(X) play an important role in many number-theoretic problems, for example Waring's problem and bounds on the zeta function.

There is an asymptotic formula: the first term is due to Rogovskaya, the second to Blomer and Brüdern.

Examples

			The system of equations have trivial solutions in which {y_1,y_2,y_3} is a permutation of {x_1,x_2,x_3}. The first nontrivial solutions are in the case J_{3,2}(5), where there are 18 solutions from permutations of {x_1,x_2,x_3} = {1,4,4}, {y_1,y_2,y_3} = {2,2,5}.
		

References

• Rogovskaya, N. N., An asymptotic formula for the number of solutions of a system of equations, Diophantine Approximations, Part II, Moskov, Gos. Univ., Moscow, 1986, pp. 78-84.

Links

• Giovanni Resta, Table of n, a(n) for n = 1..500
• V. Blomer and J. Brüdern, The number of integer points on Vinogradov's quadric, Monatsh. Math. 160 (2010) 91-107.
• Trevor D. Wooley, Translation invariance, exponential sums, and Waring's problem, arXiv:1404.3508 [math.NT], 2014.

Programs

Mathematica

J32[X_] := Module[{T, n, count, P, S, PS, long, K, L, m},
   T = Table[n, {n, 1, X}];
   count = 0;
   P = Tuples[T, 3];
   For[S = 3, S <= 3 X, S++,
    PS[S] = Select[P, Total[#] == S &]];
   For[S = 3, S <= 3 X, S++,
    long = Length[PS[S]];
    For[n = 1, n <= long, n++,
     K = PS[S][[n]];
     For[m = 1, m <= long, m++,
      L = PS[S][[m]];
      If[Total[K^2] == Total[L^2], count = count + 1]]];
    ];
   count];
Table[J32[n], {n, 1, 12}]
(* or *)
a[n_] := Sum[Block[{p,w,e}, p = IntegerPartitions[s, {3}, Range@ n]; w = Length /@ Permutations /@ p; e = (Plus @@ Last /@ #) & /@ GatherBy[ Transpose@ {Plus @@@ (p^2), w}, First]; Total[e^2]], {s, 3, 3 n}]; Array[a, 50] (* faster, Giovanni Resta, Mar 12 2017 *)

Formula

a(n) ~ (18/Pi^2)(n^3*log n) + (3/Pi^2)*(12*C - 6zeta'(2)/zeta(2) - 5)*n^3 + O(n^(5/2)log n), where C is Euler's constant.