|
%I #15 Oct 07 2020 05:44:07
%S 1,2,2,1,1,1,1,1,2,3,2,2,2,4,2,1,5,5,2,1,2,2,2,1,4,5,2,1,4,7,1,2,5,3,
%T 2,1,3,5,4,2,8,5,1,3,5,6,1,2,4,8,5,4,2,4,4,2,6,5,5,2,1,6,4,1,8,9,6,2,
%U 3,4,1,2,6,8,5,4,5,8,2,1
%N Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 3*y + 5*z a positive square, where x,y,z,w are nonnegative integers such that 3*x or y or z is a square.
%C Conjecture: a(n) > 0 for all n = 1,2,3,....
%C This is stronger than the author's 1-3-5 conjecture in A271518. See also A300751 for a similar conjecture stronger than the 1-3-5 conjecture.
%C In 2020, A. Machiavelo, R. Reis and N. Tsopanidis verified a(n) > 0 for n up to 1.05*10^11. - _Zhi-Wei Sun_, Oct 06 2020
%H Zhi-Wei Sun, <a href="/A300752/b300752.txt">Table of n, a(n) for n = 1..10000</a>
%H António Machiavelo, Rogério Reis, and Nikolaos Tsopanidis, <a href="https://arxiv.org/abs/2005.13526">Report on Zhi-Wei Sun's "1-3-5 conjecture" and some of its refinements</a>, arXiv:2005.13526 [math.NT], 2020.
%H 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 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 a(71) = 1 since 71 = 3^2 + 1^2 + 6^2 + 5^2 with 1 = 1^2 and 3 + 3*1 + 5*6 = 6^2.
%e a(248) = 1 since 248 = 10^2 + 2^2 + 0^2 + 12^2 with 0 = 0^2 and 10 + 3*2 + 5*0 = 4^2.
%e a(263) = 1 since 263 = 13^2 + 2^2 + 9^2 + 3^2 with 9 = 3^2 and 13 + 3*2 + 5*9 = 8^2.
%e a(808) = 1 since 808 = 12^2 + 14^2 + 18^2 + 12^2 with 3*12 = 6^2 and 12 + 3*14 + 5*18 = 12^2.
%e a(1288) = 1 since 1288 = 12^2 + 18^2 + 26^2 + 12^2 with 3*12 = 6^2 and 12 + 3*18 + 5*26 = 14^2.
%e a(3544) = 1 since 3544 = 14^2 + 34^2 + 16^2 + 44^2 with 16 = 4^2 and 14 + 3*34 + 5*16 = 14^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t tab={};Do[r=0;Do[If[(SQ[3(m^2-3y-5z)]||SQ[y]||SQ[z])&&SQ[n-(m^2-3y-5z)^2-y^2-z^2],r=r+1],{m,1,(35n)^(1/4)},{y,0,Min[m^2/3,Sqrt[n]]},{z,0,Min[(m^2-3y)/5,Sqrt[n-y^2]]}];tab=Append[tab,r],{n,1,80}];Print[tab]
%Y Cf. A000118, A000290, A271518, A281976, A282542, A300666, A300667, A300708, A300712, A300751.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Mar 11 2018
|
