cp's OEIS Frontend

A301375 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w a positive integer such that x*(y+3*z) is a cube or half a cube and y <= z <= w if x = 0.

%I A301375 #8 Mar 20 2018 00:12:11
%S A301375 1,2,2,2,3,2,2,2,3,5,2,3,4,1,1,1,5,7,3,3,5,3,1,4,6,7,4,3,6,1,4,2,6,7,
%T A301375 1,5,4,4,2,5,5,4,5,2,8,4,2,1,4,7,5,7,6,4,3,3,4,7,2,1,5,3,2,2,7,10,6,4,
%U A301375 6,3,4,5,9,8,5,4,2,6,2,3
%N A301375 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w a positive integer such that x*(y+3*z) is a cube or half a cube and y <= z <= w if x = 0.
%C A301375 Conjecture: a(n) > 0 for all n > 0.
%C A301375 We have verified this for all n = 1..3*10^6.
%H A301375 Zhi-Wei Sun, <a href="/A301375/b301375.txt">Table of n, a(n) for n = 1..10000</a>
%H A301375 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167-190.
%H A301375 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017-2018.
%e A301375 a(14) = 1 since 14 = 0^2 + 1^2 + 2^2 + 3^2 with 0*(1+3*2) = 0^3.
%e A301375 a(15) = 1 since 15 = 2^2 + 1^2 + 1^2 + 3^2 with 2*(1+3*1) = 2^3.
%e A301375 a(16) = 1 since 16 = 0^2 + 0^2 + 0^2 + 4^2 with 0*(0+3*0) = 0^3.
%e A301375 a(60) = 1 since 60 = 4^2 + 2^2 + 2^2 + 6^2 with 4*(2+3*2) = 4^3/2.
%e A301375 a(92) = 1 since 92 = 6^2 + 6^2 + 4^2 + 2^2 with 6*(6+3*4) = 6^3/2.
%e A301375 a(240) = 1 since 240 = 2^2 + 14^2 + 6^2 + 2^2 with 2*(14+3*6) = 4^3.
%e A301375 a(807) = 1 since 807 = 1^2 + 21^2 + 2^2 + 19^2 with 1*(21+3*2) = 3^3.
%t A301375 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A301375 CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
%t A301375 QQ[n_]:=QQ[n]=CQ[n]||CQ[2n];
%t A301375 tab={};Do[r=0;Do[If[QQ[x(y+3z)]&&SQ[n-x^2-y^2-z^2],r=r+1],{x,0,Sqrt[n-1]},{y,0,Sqrt[If[x==0,n/3,n-1-x^2]]},{z,If[x==0,y,0],Sqrt[If[x==0,(n-x^2-y^2)/2,n-1-x^2-y^2]]}]; tab=Append[tab,r],{n,1,80}];Print[tab]
%Y A301375 Cf. A000118, A000290, A271510, A271513, A271518, A281976, A300666, A300667, A300708, A300712, A300751, A300752, A300791, A300792, A300844, A300908, A301303, A301304, A301314.
%K A301375 nonn
%O A301375 1,2
%A A301375 _Zhi-Wei Sun_, Mar 19 2018