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

Author

Zhi-Wei Sun, Feb 04 2022

Keywords

• nonn

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

Examples

			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.
		

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.

Crossrefs

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

Programs

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]

Formula

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