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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, 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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 17 16:21 EDT 2022. Contains 356189 sequences. (Running on oeis4.)