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Number of ways to write n as w^2 + 2*x^2 + 3*y^4 + 4*z^4, where w,x,y,z are nonnegative integers.
6

%I #32 Jan 26 2022 13:09:30

%S 1,1,1,2,3,2,3,3,3,4,2,3,4,3,1,3,4,1,3,3,2,3,4,2,2,4,2,3,2,2,2,3,2,2,

%T 3,1,5,3,2,3,4,3,1,3,2,2,1,2,4,2,3,5,4,3,7,4,3,7,5,2,4,6,1,2,6,2,6,5,

%U 5,4,8,5,5,7,2,8,8,2,2,7,4,6,5,4,7,8,7,1,7,6,3,5,4,3,2,2,5,4,3,7,8

%N Number of ways to write n as w^2 + 2*x^2 + 3*y^4 + 4*z^4, where w,x,y,z are nonnegative integers.

%C 1-2-3-4 Conjecture: a(n) > 0 except for n = 158.

%C This has been verified for n up to 10^8.

%C It seems that a(n) = 1 only for n = 0, 1, 2, 14, 17, 35, 42, 46, 62, 87, 119, 122, 168, 189, 206, 234, 237, 302, 317, 398, 545, 1037, 1437, 4254.

%C See also A347865 and A350857 for similar conjectures.

%H Zhi-Wei Sun, <a href="/A346643/b346643.txt">Table of n, a(n) for n = 0..10000</a>

%H Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/179b.pdf">New conjectures on representations of integers (I)</a>, Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97-120.

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/2010.05775">Sums of four rational squares with certain restrictions</a>, arXiv:2010.05775 [math.NT], 2020-2022.

%e a(46) = 1 with 46 = 5^2 + 2*3^2 + 3*1^4 + 4*0^4.

%e a(119) = 1 with 119 = 7^2 + 2*3^2 + 3*2^4 + 4*1^4.

%e a(398) = 1 with 398 = 13^2 + 2*9^2 + 3*1^4 + 4*2^4.

%e a(545) = 1 with 545 = 19^2 + 2*6^2 + 3*2^4 + 4*2^4.

%e a(1037) = 1 with 1037 = 31^2 + 2*6^2 + 3*0^4 + 4*1^4.

%e a(1437) = 1 with 1437 = 9^2 + 2*26^2 + 3*0^4 + 4*1^4.

%e a(4254) = 1 with 4254 = 45^2 + 2*31^2 + 3*3^4 + 4*2^4.

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

%t tab={};Do[r=0;Do[If[SQ[n-4x^4-3y^4-2z^2],r=r+1],{x,0,(n/4)^(1/4)},{y,0,((n-4x^4)/3)^(1/4)},{z,0,Sqrt[(n-4x^4-3y^4)/2]}];tab=Append[tab,r],{n,0,100}];Print[tab]

%Y Cf. A000290, A000583, A347865, A350857.

%K nonn

%O 0,4

%A _Zhi-Wei Sun_, Jan 24 2022