cp's OEIS Frontend

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

%I A281391 #15 Mar 12 2017 21:22:37
%S A281391 1,20,93,256,563,1032,1771,2744,4077,5788,7985,10560,13855,17600,
%T A281391 22047,27304,33425,40140,47989,56504,66315,77296,89411,102336,117061,
%U A281391 132956,150201,168904,189479,211080,235111,260240,287385,316420,347237
%N A281391 Vinogradov's number J_{3,2}(n).
%C A281391 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.
%C A281391 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.
%C A281391 There is an asymptotic formula: the first term is due to Rogovskaya, the second to Blomer and Brüdern.
%D A281391 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.
%H A281391 Giovanni Resta, <a href="/A281391/b281391.txt">Table of n, a(n) for n = 1..500</a>
%H A281391 V. Blomer and J. Brüdern, <a href="http://dx.doi.org/10.1007/s00605-008-0085-8">The number of integer points on Vinogradov's quadric</a>, Monatsh. Math. 160 (2010) 91-107.
%H A281391 Trevor D. Wooley, <a href="https://arxiv.org/abs/1404.3508">Translation invariance, exponential sums, and Waring's problem</a>, arXiv:1404.3508 [math.NT], 2014.
%F A281391 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.
%e A281391 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}.
%t A281391 J32[X_] := Module[{T, n, count, P, S, PS, long, K, L, m},
%t A281391    T = Table[n, {n, 1, X}];
%t A281391    count = 0;
%t A281391    P = Tuples[T, 3];
%t A281391    For[S = 3, S <= 3 X, S++,
%t A281391     PS[S] = Select[P, Total[#] == S &]];
%t A281391    For[S = 3, S <= 3 X, S++,
%t A281391     long = Length[PS[S]];
%t A281391     For[n = 1, n <= long, n++,
%t A281391      K = PS[S][[n]];
%t A281391      For[m = 1, m <= long, m++,
%t A281391       L = PS[S][[m]];
%t A281391       If[Total[K^2] == Total[L^2], count = count + 1]]];
%t A281391     ];
%t A281391    count];
%t A281391 Table[J32[n], {n, 1, 12}]
%t A281391 (* or *)
%t A281391 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 *)
%K A281391 nonn
%O A281391 1,2
%A A281391 _Juan Arias-de-Reyna_, Jan 21 2017