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%I #19 Jan 26 2022 12:51:16
%S 1,2,2,3,4,3,3,3,2,4,3,2,5,3,1,2,3,3,4,6,5,4,6,3,2,6,2,5,7,1,3,3,2,4,
%T 5,4,6,7,4,3,3,4,2,4,4,2,3,2,4,6,5,7,10,4,7,7,1,9,6,3,7,3,2,2,4,5,7,
%U 11,6,4,9,3,5,11,2,7,10,2,2,2,4,8,12,7,9,10,7,6,5,7,6,7,8,5,1,2,4,10,7,11,15
%N Number of ways to write n as w^2 + 2*x^2 + y^4 + 3*z^4, where w,x,y,z are nonnegative integers.
%C Conjecture 1: a(n) > 0 except for n = 744.
%C This has been verified for n up to 10^8.
%C It seems that a(n) = 1 only for n = 0, 14, 29, 56, 94, 110, 158, 159, 224, 239, 296, 464, 589, 1214, 1454, 1709.
%C Conjecture 2: For any positive odd integer a, all sufficiently large integers can be written as a*w^4 + 2*x^4 + (2*y)^2 + z^2 with w,x,y,z integers. If M(a) denotes the largest integer not of the form a*w^4 + 2*x^4 + (2*y)^2 + z^2 (with w,x,y,z integers), then M(1) = 255, M(3) = 303, M(5) = 497, M(7) = 3182, M(9) = 4748, M(11) = 5662, M(13) = 5982, M(15) = 10526, M(17) = 4028 and M(19) = 11934.
%C Conjecture 3: Let E(a,b,c) be the set of nonnegative integers not of the form w^2 + a*x^2 + b*y^4 + c*z^4 with w,x,y,z integers. Then E(1,2,4) = {135, 190, 510}, E(1,2,5) = {35, 254, 334}, E(2,1,4) = {190, 270, 590} and E(2,3,7) = {94, 490, 983} and E(3,1,2) = {56, 168, 378}.
%C See also A346643 and A350857 for similar conjectures.
%H Zhi-Wei Sun, <a href="/A347865/b347865.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(14) = 1 with 14 = 3^2 + 2*1^2 + 0^4 + 3*1^4.
%e a(158) = 1 with 158 = 11^2 + 2*3^2 + 2^4 + 3*1^4.
%e a(589) = 1 with 589 = 14^2 + 2*14^2 + 1^4 + 3*0^4.
%e a(1214) = 1 with 1214 = 27^2 + 2*11^2 + 0^4 + 3*3^4.
%e a(1454) = 1 with 1454 = 27^2 + 2*19^2 + 0^4 + 3*1^4.
%e a(1709) = 1 with 1709 = 29^2 + 2*0^2 + 5^4 + 3*3^4.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t tab={};Do[r=0;Do[If[SQ[n-3x^4-y^4-2z^2],r=r+1],{x,0,(n/3)^(1/4)},{y,0,(n-3x^4)^(1/4)},
%t {z,0,Sqrt[(n-3x^4-y^4)/2]}];tab=Append[tab,r],{n,0,100}];Print[tab]
%Y Cf. A000290, A000583, A346643, A350857.
%K nonn
%O 0,2
%A _Zhi-Wei Sun_, Jan 24 2022
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