A352259 - OEIS

%I #17 Mar 11 2022 12:29:36

%S 1,2,2,3,4,3,2,3,3,3,2,3,6,4,3,2,2,5,5,5,4,3,4,2,1,5,5,4,6,5,3,3,4,5,

%T 4,5,7,5,4,5,4,3,3,3,4,3,3,5,6,7,6,5,7,6,4,4,4,7,5,4,4,3,7,5,5,6,6,10,

%U 8,3,3,4,5,8,4,9,13,12,8,2,7,10,9,10,9,7,5,3,3,8,5,10,10,6,7,8,6,10,9,11,10

%N Number of ways to write n as w^6 + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.

%C Conjecture 1: (i) a(n) > 0 for every n = 0,1,2,.... Moreover, 106, 744, 5469 and 331269 are the only nonnegative integers not in the set {w + x^2 + 2*y^2 + 3*z^2 + x*y*z: w = 0,1; x,y,z = 0,1,2,...}.

%C (ii) Let k be one of 4, 5, 6, 7. Then each n = 0,1,2,... can be written as 10*w^k + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.

%C (iii) Let c be among 1, 3, 4, 6, 7, and let k be 4 or 5. Then every n = 0,1,2,... can be written as c*w^k + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.

%C (iv) Each n = 0,1,2,... can be written as 9*w^4 + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w,x,y,z are nonnegative integers.

%C Conjecture 2: Every n = 0,1,2,... can be written as 2*w^4 + 3*x^2 + y^2 + z^2 + x*y*z, where w,x,y,z are nonnegative integers.

%C We have verified Conjectures 1 and 2 for all n <= 10^5.

%H Zhi-Wei Sun, <a href="/A352259/b352259.txt">Table of n, a(n) for n = 0..10000</a>

%e a(24) = 1 with 24 = 0^6 + 4^2 + 2*2^2 + 3*0^2 + 4*2*0.

%e a(106) = 1 with 106 = 2^6 + 1^2 + 2*2^2 + 3*3^2 + 1*2*3.

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

%t tab={};Do[r=0;Do[If[SQ[4(n-w^6-2y^2-3z^2)+y^2*z^2],r=r+1],{w,0,n^(1/6)},{z,0,Sqrt[(n-w^6)/3]},{y,0,Sqrt[(n-w^6-3z^2)/2]}];tab=Append[tab,r],{n,0,100}];Print[tab]

%Y Cf. A000290, A000583, A000584, A001014, A351723, A351617, A351902, A352286.

%K nonn

%O 0,2

%A _Zhi-Wei Sun_, Mar 10 2022