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.

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, 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, 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

Offset

0,6

Comments

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).

Aside from a(96) = 7 and a(850) = 8, a(n) <= 6 for n <= 10^6. - Charles R Greathouse IV, Feb 10 2022

References

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.)

Links

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

H. W. Richmond, On analogues of Waring's problem for rational numbers, Proceedings of the London Mathematical Society, s2-21 (1923), pp. 401-409.

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.

Formula

a(n) <= A351199(n)^2. - Charles R Greathouse IV, Feb 05 2022

Example

a(5) = 3 with 3^6*5 = 2^6 + 5^3 + 12^3 + 12^3.
a(12) = 5 with 5^6*12 = 3^6 + 19^3 + 34^3 + 52^3.
a(22) = 2 with 2^6*22 = 1^6 + 4^3 + 7^3 + 10^3.
a(31) = 6 with 6^6*31 = 0^6 + 4^3 + 15^3 + 113^3.
a(96) = 7 with 7^6*96 = 0^6 + 2^3 + 38^3 + 224^3.
a(101) = 4 with 4^6*101 = 3^6 + 22^3 + 39^3 + 70^3.
a(850) = 8 with 8^6*850 = 5^6 + 508^3 + 442^3 + 175^3.

Mathematica

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
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)}];
m=m+1; Goto[bb]; Label[aa], {n, 0, 100}]; Print[tab]

Crossrefs

Cf. A000578, A001014, A004825, A350714, A351199.

Sequence in context: A010615 A114865 A096196 * A379946 A076824 A103728

Adjacent sequences: A351176 A351177 A351178 * A351180 A351181 A351182

Keyword

nonn

Author

Zhi-Wei Sun, Feb 04 2022

Status

approved