A351199 Least positive integer m such that m^3*n = x^3 + y^3 + z^3 for some nonnegative integers x,y,z.

1, 1, 1, 1, 15, 6, 5, 3, 1, 1, 1, 11, 39, 3, 3, 3, 1, 1, 4, 2, 2, 3, 18, 6, 1, 22, 28, 1, 1, 1, 29, 15, 15, 21, 3, 1, 1, 7, 7, 25, 3, 12, 6, 1, 2, 7, 2, 7, 5, 21, 6, 2, 25, 5, 1, 1, 3, 3, 45, 132, 6, 45, 1, 3, 1, 1, 1, 171, 6, 9, 2, 3, 1, 1, 54, 21, 18, 3, 13, 32, 1, 1, 10, 2, 7, 9, 3, 3, 6, 3, 11, 1, 1, 63, 3, 30, 21, 5, 4, 1, 12

Offset

0,5

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).

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 (terms 0..600 from Zhi-Wei Sun)

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) >= sqrt(A351179(n)).

Example

a(4) = 15 with 15^3*4 = 12^3 + 17^3 + 19^3.
a(212) = 216 with 216^3*212 = 82^3 + 161^3 + 1287^3.
a(446) = 228 with 228^3*446 = 929^3 + 1287^3 + 1330^3.

Mathematica

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
tab={}; Do[m=1; Label[bb]; k=m^3; Do[If[CQ[k*n-x^3-y^3], tab=Append[tab, m]; Goto[aa]],  {x, 0, ((k*n)/3)^(1/3)}, {y, x, ((k*n-x^3)/2)^(1/3)}];
m=m+1; Goto[bb]; Label[aa], {n, 0, 100}]

Prog

(PARI) T=thueinit('x^3+1);
has2(n)=n==0 || #select(v->min(v[1], v[2])>=0, thue(T, n))>0
has3(n)=forstep(k=sqrtnint(n, 3), sqrtnint(n\3, 3), -1, if(has2(n-k^3), return(1))); 0
a(n)=my(m=1); while(!has3(m^3*n), m++); m \\ Charles R Greathouse IV, Feb 05 2022

Crossrefs

Cf. A000578, A350714, A351179.

Sequence in context: A292307 A064107 A116907 * A277384 A040214 A364351

Adjacent sequences: A351196 A351197 A351198 * A351200 A351201 A351202

Keyword

nonn

Author

Zhi-Wei Sun, Feb 04 2022

Status

approved