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%I #16 Feb 02 2022 15:50:36
%S 1,2,2,1,1,3,3,2,2,3,4,2,0,2,3,2,3,3,4,2,2,4,3,3,2,5,6,1,0,3,4,4,3,2,
%T 4,2,2,4,3,2,2,6,4,1,0,3,5,2,1,1,6,3,2,4,2,4,3,3,4,2,0,3,2,1,2,4,6,1,
%U 2,3,4,4,1,4,5,1,0,2,2,3,4,7,6,3,2,7,9,3,4,6,9,6,0,2,5,4,5,6,7,4,4
%N Number of ways to write n as 5*w^4 + x^4 + y^2 + z^2, where w,x,y,z are nonnegative integers with y <= z.
%C Conjecture: a(n) = 0 only for n == 12 (mod 16).
%C This has been verified for n up to 10^8.
%C Now we show that a(n) = 0 whenever n == 12 (mod 16). If 16*q + 12 = 5*w^4 + x^4 + y^2 + z^2 with q,w,x,y,z integers, then the equality modulo 8 yields that w,x,y,z are all even, hence 4*q + 3 == 20*(w/2)^4 + 4*(x/2)^4 + (y/2)^2 + (z/2)^2 and thus (y/2)^2 + (z/2)^2 == 3 (mod 4) which is impossible.
%C It seems that a(n) = 1 only for n = 0, 3, 4, 27, 43, 48, 49, 63, 67, 72, 75, 192, 215, 303, 1092.
%H Zhi-Wei Sun, <a href="/A348890/b348890.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(192) = 1 with 192 = 5*1^4 + 3^4 + 5^2 + 9^2.
%e a(215) = 1 with 215 = 5*1^4 + 2^4 + 5^2 + 13^2.
%e a(303) = 1 with 303 = 5*1^4 + 0^4 + 3^2 + 17^2.
%e a(1092) = 1 with 1092 = 5*0^4 + 2^4 + 20^2 + 26^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t tab={};Do[r=0;Do[If[SQ[n-5x^4-y^4-z^2],r=r+1],{x,0,(n/5)^(1/4)},{y,0,(n-5x^4)^(1/4)},
%t {z,0,Sqrt[(n-5x^4-y^4)/2]}];tab=Append[tab,r],{n,0,100}];Print[tab]
%Y Cf. A000290, A000583, A214891, A346643, A347865, A350857.
%K nonn
%O 0,2
%A _Zhi-Wei Sun_, Jan 28 2022
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