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

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, 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, 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

Offset: 0

Author

Zhi-Wei Sun, Feb 05 2022

Keywords

• nonn

Comments

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.

Examples

			a(6) = 1 with 1^6*6 = 1^6 + 1^3 + 2^2.
a(7) = 38 with 38^6*7 = 42^6 + 1935^3 + 91337^2.
a(21) = 30 with 30^6*21 = 26^6 + 2399^3 + 34545^2.
a(22) = 14 with 14^6*22 = 0^6 + 447^3 + 8737^2.
a(96) = 26 with 26^6*96 = 21^6 + 2711^3 + 98212^2.
a(1120) = 38 with 38^6*1120 = 69^6 + 11499^3 + 1320550^2.
a(2091) = 58 with 58^6*2091 = 161^6 + 39043^3 + 1633994^2.
a(3855) = 51 with 51^6*3855 = 34^6 + 40775^3 + 199008^2.
a(3991) = 45 with 45^6*3991 = 74^6 + 3715^3 + 5738018^2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..4000
• Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no. 2, 97-120.
• Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.

Crossrefs

Cf. A000290, A000578, A001014, A350714.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
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)}];
m=m+1;Goto[bb];Label[aa],{n,0,100}];Print[tab]