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%I #17 May 25 2023 08:10:33
%S 1,2,2,2,2,2,2,2,3,4,3,2,3,3,2,1,2,4,3,4,3,2,2,3,3,3,3,4,5,2,3,2,3,5,
%T 4,4,5,3,4,3,2,3,2,2,5,5,4,2,2,5,3,5,5,3,5,5,2,3,3,4,4,2,2,4,4,6,3,5,
%U 4,2,3,4,5,5,4,4,5,5,5,1,5
%N Number of ordered ways to write n as x^3 + 2*y^2 + k*z^2, where x,y,z are nonnegative integers, k is 1 or 5, and k = 1 if z = 0.
%C Conjecture 1: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 15, 79, 120, 218, 399, 454, 622, 725, 3240.
%C We have verified that a(n) > 0 for all n = 0..10^7.
%C Conjecture 2: For any positive integers a, b, c and integers i, j, k greater than one, there are infinitely many positive integers not in the set {a*x^i + b*y^j + c*z^k: x,y,z = 0,1,2,...}. - _Zhi-Wei Sun_, May 24 2023
%H Zhi-Wei Sun, <a href="/A275150/b275150.txt">Table of n, a(n) for n = 0..10000</a>
%H G. Doyle and K. S. Williams, <a href="http://math.colgate.edu/~integers/r41/r41.Abstract.html">A positive-definite ternary quadratic form does not represent all positive integers</a>, Integers 17 (2017), #A41, 19pp (electronic).
%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(0) = 1 since 0 = 0^3 + 2*0^2 + 0^2.
%e a(15) = 1 since 15 = 2^3 + 2*1^2 + 5*1^2.
%e a(79) = 1 since 79 = 3^3 + 2*4^2 + 5*2^2.
%e a(120) = 1 since 120 = 2^3 + 2*4^2 + 5*4^2.
%e a(218) = 1 since 218 = 6^3 + 2*1^2 + 0^2.
%e a(399) = 1 since 399 = 5^3 + 2*3^2 + 16^2.
%e a(454) = 1 since 454 = 0^3 + 2*15^2 + 2^2.
%e a(622) = 1 since 622 = 2^3 + 2*17^2 + 6^2.
%e a(725) = 1 since 725 = 5^3 + 2*10^2 + 20^2.
%e a(3240) = 1 since 3240 = 7^3 + 2*38^2 + 3^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t TQ[n_]:=TQ[n]=SQ[n]||SQ[n/5]
%t Do[r=0;Do[If[TQ[n-x^3-2*y^2],r=r+1],{x,0,n^(1/3)},{y,0,Sqrt[(n-x^3)/2]}];Print[n," ",r];Continue,{n,0,80}]
%Y Cf. A000290, A000578, A262813, A262941, A262954, A270488, A274274, A275083.
%K nonn
%O 0,2
%A _Zhi-Wei Sun_, Jul 17 2016
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