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%I #17 Aug 28 2022 13:01:08
%S 1,2,3,4,4,4,5,6,5,4,3,2,3,4,3,2,3,3,4,4,4,4,5,5,4,4,3,3,3,2,2,3,4,5,
%T 5,5,5,5,5,5,3,1,3,5,4,4,4,3,5,6,4,2,3,4,3,2,2,4,5,4,4,4,4,5,5,4,4,6,
%U 5,3,2,2,5,6,5,5,5,5,7,8,4,2,4,5,4,5,5,6,7,6,6,5,6,8,9,8,6,7,5,3,3
%N Number of ways to write n as w^8 + x^4 + 2*y^4 + 4*z^4 + t*(t+1), where w, x, y, z, and t are nonnegative integers.
%C Conjecture: a(n) > 0 for all n = 0,1,2,....
%C This has been verified for all n = 0..10^8.
%C It seems that a(n) = 1 only for n = 0, 41, 131, 141, 145, 225, 251, 297, 591, 621, 916, 1021, 1241, 1431, 2025, 4691.
%H Zhi-Wei Sun, <a href="/A349246/b349246.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.
%e a(145) = 1 with 145 = 0^8 + 3^4 + 2*0^4 + 4*2^4 + 0*1.
%e a(225) = 1 with 225 = 1^8 + 2^4 + 2*3^4 + 4*1^4 + 6*7.
%e a(916) = 1 with 916 = 2^8 + 2^4 + 2*4^4 + 4*0^4 + 11*12.
%e a(1021) = 1 with 1021 = 0^8 + 5^4 + 2*0^4 + 4*3^4 + 8*9.
%e a(1241) = 1 with 1241 = 0^8 + 5^4 + 2*0^4 + 4*2^4 + 23*24.
%e a(1431) = 1 with 1431 = 1^8 + 6^4 + 2*1^4 + 4*0^4 + 11*12.
%e a(2025) = 1 with 2025 = 2^8 + 3^4 + 2*2^4 + 4*3^4 + 36*37.
%e a(4691) = 1 with 4691 = 2^8 + 3^4 + 2*0^4 + 4*2^4 + 65*66.
%t QQ[n_]:=QQ[n]=IntegerQ[Sqrt[4n+1]];
%t tab={};Do[r=0;Do[If[QQ[n-w^8-4z^4-2y^4-x^4],r=r+1],{w,0,n^(1/8)},{z,0,((n-w^8)/4)^(1/4)},{y,0,((n-w^8-4z^4)/2)^(1/4)},{x,0,(n-w^8-4z^4-2y^4)^(1/4)}];tab=Append[tab,r],{n,0,100}];Print[tab]
%Y Cf. A000583, A001016, A002378.
%K nonn
%O 0,2
%A _Zhi-Wei Sun_, Mar 26 2022
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