%I #36 Mar 11 2023 07:55:55
%S 1,1,1,1,1,1,1,2,1,1,1,1,1,1,2,3,1,1,1,2,1,1,3,2,1,1,1,1,1,1,3,1,1,1,
%T 1,2,1,1,1,2,1,1,1,1,1,1,2,1,2,1,1,1,1,1,2,2,2,1,1,1,1,2,2,1,1,1,1,2,
%U 1,1,2,3,1,1,1,4,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,2,2,1,1,1,2,1
%N Least positive integer m such that m^12*n = x^4 + y^3 + z^2 for some nonnegative integers x,y,z.
%C 4-3-2 Conjecture: a(n) exists for any nonnegative integer n. Equivalently, each nonnegative rational number can be written x^4 + y^3 + z^2 with x,y,z nonnegative rational numbers.
%C Note that m/n = (m*n^11)/n^12 for any positive integers m and n.
%C a(n) <= 4 for n <= 40000 with the only exception a(23710) = 5.
%C a(n) <= 4 for n = 77000..100000, and a(n) = 4 for n = 78367, 79479, 83494, 84694, 85979, 86822, 87395, 87814, 90047, 90278, 92891, 93715.
%C _Qing-Hu Hou_ verified a(n) <= 4 for 40000 < n < 77000. - _Zhi-Wei Sun_ Feb 04 2022
%C a(n) <= 5 for 10^5 < n <= 2*10^5, and a(n)=5 for n=107206, 117615, and 148079. - _Qing-Hu Hou_, Feb 05 2022
%H Zhi-Wei Sun, <a href="/A350714/b350714.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.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/2010.05775">Sums of four rational squares with certain restrictions</a>, arXiv:2010.05775 [math.NT], 2020-2022.
%e a(6) = 1 with 1^12*6 = 1^4 + 1^3 + 2^2.
%e a(7) = 2 with 2^12*7 = 2^4 + 15^3 + 159^2.
%e a(75) = 4 with 4^12*75 = 122^4 + 1007^3 + 3951^2.
%e a(1140) = 3 with 3^12*1140 = 0^4 + 531^3 + 21357^2.
%e a(23710) = 5 with 5^12*23710 = 217^4 + 17897^3 + 232166^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t tab={};Do[m=1;Label[bb];k=m^12;Do[If[SQ[k*n-x^4-y^3],tab=Append[tab,m];Goto[aa]],{x,0,(k*n)^(1/4)},{y,0,(k*n-x^4)^(1/3)}];m=m+1;Goto[bb];Label[aa],{n,0,100}];Print[tab]
%Y Cf. A000290, A000578, A000583, A280356.
%K nonn
%O 0,8
%A _Zhi-Wei Sun_, Feb 02 2022