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
Keywords
Examples
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.
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.
Programs
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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}] -
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
Formula
a(n) >= sqrt(A351179(n)).
Comments