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A351179 Least positive integer m such that m^6*n = w^6 + x^3 + y^3 + z^3 for some nonnegative integers w,x,y,z.

%I A351179 #21 Feb 10 2022 08:33:28
%S A351179 1,1,1,1,1,3,5,3,1,1,1,1,5,3,3,3,1,1,1,2,2,2,2,3,1,1,3,1,1,1,1,6,3,3,
%T A351179 3,1,1,1,5,3,3,3,3,1,1,2,2,2,2,3,3,2,2,2,1,1,1,3,3,3,3,3,1,1,1,1,1,1,
%U A351179 6,3,2,2,1,1,1,3,3,3,3,3,1,1,1,2,2,2,3,3,1,2,3,1,1,1,3,3,7,3,2,1,1
%N A351179 Least positive integer m such that m^6*n = w^6 + x^3 + y^3 + z^3 for some nonnegative integers w,x,y,z.
%C A351179 a(n) always exists, because any positive rational number can be written as a sum of three cubes of positive rational numbers (see Richmond reference).
%C A351179 Aside from a(96) = 7 and a(850) = 8, a(n) <= 6 for n <= 10^6. - _Charles R Greathouse IV_, Feb 10 2022
%D A351179 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th Edition, Oxford Univ. Press, 1960. (See Theorem 234 on page 197.)
%H A351179 Charles R Greathouse IV, <a href="/A351179/b351179.txt">Table of n, a(n) for n = 0..10000</a>
%H A351179 H. W. Richmond, <a href="https://doi.org/10.1112/plms/s2-21.1.401">On analogues of Waring's problem for rational numbers</a>, Proceedings of the London Mathematical Society, s2-21 (1923), pp. 401-409.
%H A351179 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 A351179 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.
%F A351179 a(n) <= A351199(n)^2. - _Charles R Greathouse IV_, Feb 05 2022
%e A351179 a(5) = 3 with 3^6*5 = 2^6 + 5^3 + 12^3 + 12^3.
%e A351179 a(12) = 5 with 5^6*12 = 3^6 + 19^3 + 34^3 + 52^3.
%e A351179 a(22) = 2 with 2^6*22 = 1^6 + 4^3 + 7^3 + 10^3.
%e A351179 a(31) = 6 with 6^6*31 = 0^6 + 4^3 + 15^3 + 113^3.
%e A351179 a(96) = 7 with 7^6*96 = 0^6 + 2^3 + 38^3 + 224^3.
%e A351179 a(101) = 4 with 4^6*101 = 3^6 + 22^3 + 39^3 + 70^3.
%e A351179 a(850) = 8 with 8^6*850 = 5^6 + 508^3 + 442^3 + 175^3.
%t A351179 CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
%t A351179 tab={};Do[m=1;Label[bb];k=m^6;Do[If[CQ[k*n-w^6-x^3-y^3],tab=Append[tab,m];Goto[aa]],{w,0,(k*n)^(1/6)},{x,0,((k*n-w^6)/3)^(1/3)},{y,x,((k*n-w^6-x^3)/2)^(1/3)}];
%t A351179 m=m+1;Goto[bb];Label[aa],{n,0,100}];Print[tab]
%Y A351179 Cf. A000578, A001014, A004825, A350714, A351199.
%K A351179 nonn
%O A351179 0,6
%A A351179 _Zhi-Wei Sun_, Feb 04 2022