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%I #25 Mar 11 2022 12:29:21
%S 1,1,1,2,2,1,1,2,1,2,1,4,4,3,4,1,3,5,3,5,1,5,5,1,3,4,3,6,5,5,2,4,4,3,
%T 2,8,4,5,5,5,2,4,3,5,3,5,5,5,5,7,3,5,5,4,4,3,4,8,3,8,2,6,8,3,5,4,5,10,
%U 1,5,1,4,7,4,4,7,8,11,1,3,4,5,6,7,5,6,7,7,1,5,4,10,4,7,7,4,3,7,3,8
%N Number of ways to write n as 11^w + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.
%C Conjecture: (i) a(n) > 0 for all n > 0.
%C (ii) Let c be among 3, 4, 5, 7, 8. Then each positive integer n can be written as c^w + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.
%C This has been verified for all n = 1..3*10^5.
%H Zhi-Wei Sun, <a href="/A351617/b351617.txt">Table of n, a(n) for n = 1..10000</a>
%e a(6) = 1 with 6 = 11^0 + 0^2 + 2*1^2 + 3*1^2 + 0*1*1.
%e a(24) = 1 with 24 = 11^1 + 1^2 + 2*0^2 + 3*2^2 + 1*0*2.
%e a(71) = 1 with 71 = 11^0 + 4^2 + 2*3^2 + 3*2^2 + 4*3*2.
%e a(89) = 1 with 89 = 11^0 + 4^2 + 2*6^2 + 3*0^2 + 4*6*0.
%e a(107) = 1 with 107 = 11^1 + 8^2 + 2*4^2 + 3*0^2 + 8*4*0.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t tab={};Do[r=0;Do[If[SQ[4(n-11^w-2y^2-3z^2)+y^2*z^2],r=r+1],{w,0,Log[11,n]},{z,0,Sqrt[(n-11^w)/3]},{y,0,Sqrt[(n-11^w-3z^2)/2]}];tab=Append[tab,r],{n,1,100}];Print[tab]
%Y Cf. A000290, A001014, A351723, A351902, A352259, A352286.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Mar 10 2022
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