A161883 Smallest k such that n^3 = a_1^3+...+a_k^3 and all a_i are positive integers less than n.

8, 6, 5, 7, 3, 4, 5, 3, 5, 5, 3, 4, 4, 5, 5, 5, 3, 3, 3, 4, 5, 4, 3, 3, 4, 3, 3, 3, 3, 4, 4, 4, 4, 4, 3, 4, 3, 4, 3, 3, 3, 4, 3, 3, 3, 5, 3, 4, 3, 4, 4, 3, 3, 4, 3, 3, 3, 4, 3, 5, 4, 3, 4, 4, 3, 3, 4, 3, 3, 3, 3, 4, 4, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3

Offset: 2

Author

Dmitry Kamenetsky, Jun 21 2009

Keywords

• nonn

Comments

It follows from Wieferich's result g(3) = 9 that a(n) <= 10. Theorem 2 of Bertault, Ramaré, & Zimmermann can be used to show that a(n) <= 8 (check congruence classes of cubes mod 333 with one summand of 1, 8, or 27). Probably a(2), a(3), and a(5) are the only members greater than 5 in this sequence. - Charles R Greathouse IV, Jul 30 2011

Links

• Giovanni Resta, Table of n, a(n) for n = 2..10000
• F. Bertault, O. Ramaré, and P. Zimmermann, On sums of seven cubes, Mathematics of Computation 68 (1999), pp. 1303-1310.
• Jean-Charles Meyrignac, Computing minimal equal sums of like powers
• Manfred Scheucher, Sage Script
• Eric W. Weisstein, Diophantine Equation 3rd Powers
• Eric W. Weisstein, Waring's Problem

Crossrefs

Cf. A161882, A161884, A161885.

Programs

Mathematica

f[n_, k_] := Select[PowersRepresentations[n^3, k, 3], AllTrue[#, 0<#Jean-François Alcover, Oct 03 2020 *)

PARI

A161883(n,verbose=0,m=3)={N=n^m;for(k=3,99,forvec(v=vector(k-1,i,[1,n\sqrtn(k+1-i,m)]),ispower(N-sum(i=1,k-1,v[i]^m),m,&K)&&K>0&&!if(verbose,print1("/*"n" "v"*/"))&&return(k),1))} \\ M. F. Hasler, Dec 17 2014

Extensions

More terms from M. F. Hasler, Dec 17 2014