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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 04 2022
STATUS
approved