%I #15 Mar 27 2022 10:54:31
%S 1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,3,4,3,3,3,3,3,4,3,4,3,4,3,4,3,3,4,
%T 2,4,5,4,3,4,5,4,2,3,5,4,2,3,7,3,3,4,5,3,2,4,5,3,1,4,5,3,1,5,5,4,3,5,
%U 6,3,3,3,6,3,4,6,5,2,3,3,7,6,4,6,4,5,2,5,7,7,6,7,3,6,2,6,8,4,5,6
%N Number of ways to write n as t^2 + w^4 + 2*x^4 + 4*y^4 + 8*z^4, where t is a positive integer, and w,x,y,z are nonnegative integers.
%C Conjecture: If (a,b,c) is among the triples (2,3,4), (2,3,8), (2,4,6), (2,4,7), (2,4,8) and (2,4,11), then any positive integer n can be written as t^2 + w^4 + a*x^4 + b*y^4 + c*z^4, where t is a positive integer, and w,x,y,z are nonnegative integers.
%C We have verified this for all n = 1..10^8.
%C It seems that a(n) = 1 only for n = 1, 2, 3, 59, 63, 119, 223, 1087.
%H Zhi-Wei Sun, <a href="/A350088/b350088.txt">Table of n, a(n) for n = 1..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.
%e a(1) = 1 with 1 = 1^2 + 0^4 + 2*0^4 + 4*0^4 + 8*0^4.
%e a(2) = 1 with 2 = 1^2 + 1^4 + 2*0^4 + 4*0^4 + 8*0^4.
%e a(3) = 1 with 3 = 1^2 + 0^4 + 2*1^4 + 4*0^4 + 8*0^4.
%e a(59) = 1 with 59 = 7^2 + 0^4 + 2*1^4 + 4*0^4 + 8*1^4.
%e a(63) = 1 with 63 = 7^2 + 0^4 + 2*1^4 + 4*1^4 + 8*1^4.
%e a(119) = 1 with 119 = 6^2 + 3^4 + 2*1^4 + 4*0^4 + 8*0^4.
%e a(223) = 1 with 223 = 7^2 + 0^4 + 2*3^4 + 4*1^4 + 8*1^4.
%e a(1087) = 1 with 1087 = 14^2 + 3^4 + 2*3^4 + 4*0^4 + 8*3^4.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t tab={}; Do[r=0; Do[If[SQ[n-w^4-2x^4-4*y^4-8z^4], r=r+1], {w, 0, (n-1)^(1/4)}, {x, 0, ((n-1-w^4)/2)^(1/4)}, {y, 0, ((n-1-w^4-2x^4)/4)^(1/4)},{z, 0, ((n-1-w^4-2x^4-4y^4)/8)^(1/4)}]; tab=Append[tab,r], {n, 1, 100}]; Print[tab]
%Y Cf. A000290, A000583, A349246.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Mar 26 2022