cp's OEIS Frontend

A351221 Least positive integer m such that m^6*n = x^6 + y^3 + z^2 for some nonnegative integers x,y,z.

%I A351221 #12 Feb 05 2022 23:39:23
%S A351221 1,1,1,1,1,1,1,38,1,1,1,1,1,1,18,3,1,1,1,2,8,30,14,2,1,1,1,1,1,1,4,1,
%T A351221 1,1,1,4,1,1,1,3,8,3,3,1,1,1,2,2,13,1,1,1,1,1,2,2,4,1,1,2,9,2,2,1,1,1,
%U A351221 1,3,1,1,3,3,1,1,1,2,1,1,5,2,1,1,1,1,11,9,2,3,1,1,1,1,1,3,3,1,26,1,2,2,1
%N A351221 Least positive integer m such that m^6*n = x^6 + y^3 + z^2 for some nonnegative integers x,y,z.
%C A351221 6-3-2 Conjecture: a(n) exists for any nonnegative integer n. Equivalently, each nonnegative rational number can be written as x^6 + y^3 + z^2 with x,y,z nonnegative rational numbers.
%H A351221 Zhi-Wei Sun, <a href="/A351221/b351221.txt">Table of n, a(n) for n = 0..4000</a>
%H A351221 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 A351221 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 A351221 a(6) = 1 with 1^6*6 = 1^6 + 1^3 + 2^2.
%e A351221 a(7) = 38 with 38^6*7 = 42^6 + 1935^3 + 91337^2.
%e A351221 a(21) = 30 with 30^6*21 = 26^6 + 2399^3 + 34545^2.
%e A351221 a(22) = 14 with 14^6*22 = 0^6 + 447^3 + 8737^2.
%e A351221 a(96) = 26 with 26^6*96 = 21^6 + 2711^3 + 98212^2.
%e A351221 a(1120) = 38 with 38^6*1120 = 69^6 + 11499^3 + 1320550^2.
%e A351221 a(2091) = 58 with 58^6*2091 = 161^6 + 39043^3 + 1633994^2.
%e A351221 a(3855) = 51 with 51^6*3855 = 34^6 + 40775^3 + 199008^2.
%e A351221 a(3991) = 45 with 45^6*3991 = 74^6 + 3715^3 + 5738018^2.
%t A351221 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A351221 tab={};Do[m=1;Label[bb];k=m^6;Do[If[SQ[k*n-x^6-y^3],tab=Append[tab,m];Goto[aa]],{x,0,(k*n)^(1/6)},{y,0,(k*n-x^6)^(1/3)}];
%t A351221 m=m+1;Goto[bb];Label[aa],{n,0,100}];Print[tab]
%Y A351221 Cf. A000290, A000578, A001014, A350714.
%K A351221 nonn
%O A351221 0,8
%A A351221 _Zhi-Wei Sun_, Feb 05 2022