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

0,8

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.

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.

Example

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.

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]

Crossrefs

Cf. A000290, A000578, A001014, A350714.

Sequence in context: A171407 A023930 A022072 * A154229 A225433 A225398

Adjacent sequences: A351218 A351219 A351220 * A351222 A351223 A351224

Keyword

nonn

Author

Zhi-Wei Sun, Feb 05 2022

Status

approved