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%I #19 Jan 26 2022 11:10:29
%S 1,2,2,1,2,2,2,1,2,3,3,1,2,4,1,2,5,6,4,3,4,3,4,2,4,7,4,5,5,4,2,4,6,5,
%T 4,3,6,8,2,1,8,7,6,2,4,6,2,3,5,7,6,7,10,4,1,6,4,7,4,2,5,6,6,4,7,9,5,7,
%U 5,1,3,2,8,6,4,1,6,7,3,4,6,6,7,5,1,7,2,7,3,6,5,1,3,5,5,3,7,11,4,2
%N Number of ways to write n as 16^w + x^2 + (y^2 + 23*z^4)/324, where w,x,y,z are nonnegative integers.
%C Conjecture: a(n) > 0 for all n > 0.
%C This has been verified for n up to 10^6.
%C It seems that a(n) = 1 only for n = 1, 4, 8, 12, 15, 40, 55, 70, 76, 85, 92, 104, 135, 156, 177, 192, 204, 207, 231, 279, 300, 447, 567, 1427, 1887, 4371.
%C See also A347827 for a similar conjecture.
%H Zhi-Wei Sun, <a href="/A347562/b347562.txt">Table of n, a(n) for n = 1..10000</a>
%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(15) = 1 with 15 = 16^0 + 1^2 + (62^2 + 23*2^4)/324.
%e a(156) = 1 with 156 = 16^1 + 6^2 + (139^2 + 23*5^4)/324.
%e a(300) = 1 with 300 = 16^2 + 6^2 + (27^2 + 23*3^4)/324.
%e a(1427) = 1 with 1427 = 16^1 + 35^2 + (71^2 + 23*7^4)/324.
%e a(1887) = 1 with 1887 = 16^1 + 15^2 + (729^2 + 23*3^4)/324.
%e a(4371) = 1 with 4371 = 16^1 + 63^2 + (351^2 + 23*3^4)/324.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t tab={};Do[r=0;Do[If[SQ[324(n-16^w-x^2)-23y^4],r=r+1],{w,0,Log[16,n]},{x,0,Sqrt[n-16^w]},
%t {y,0,(324(n-16^w-x^2)/23)^(1/4)}];tab=Append[tab,r],{n,1,100}];Print[tab]
%Y Cf. A000290, A000583, A001025, A347824, A347827, A350860.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Jan 23 2022
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