A252486 Smallest k such that n^6 = a_1^6+...+a_k^6 where all the a_i are positive integers less than n.

64, 36, 15, 29, 22, 21, 15, 19, 15, 17, 15, 16, 14, 15, 13, 12, 11, 11, 13, 14, 12, 13, 13, 12, 12, 12, 12, 12, 11, 11, 11, 11, 11, 13, 11, 11, 11, 10, 11, 11, 11, 11, 11, 10, 11, 11, 11, 11, 11, 11, 11, 11, 9, 11, 10, 11, 11, 11, 9, 10, 11, 11, 11, 11, 10

Offset: 2

Author

M. F. Hasler, Dec 17 2014

Keywords

• nonn

Comments

Inspired by Fermat's Last Theorem: 2 never occurs in this sequence.

No n is known for which a(n)<7, according to the MathWorld page. The values 7, 8, 9, 10 and 11 occur first at indices 1141, 251, 54, 39, 18, cf. sequence A252476.

I conjecture that the sequence is bounded by the initial term. Probably even a(3)=36, a(5)=29, a(6)=22 and some more are followed only by smaller terms.

From results on Waring's problem, it is known that all a(n) <= A002804(6) = 73, and a(n) <= 24 for all sufficiently large n. - Robert Israel, Aug 17 2015

Links

• Giovanni Resta, Table of n, a(n) for n = 2..200
• Jean-Charles Meyrignac, Computing Minimal Equal Sums Of Like Powers
• Manfred Scheucher, Sage Script
• Eric W. Weisstein, Diophantine Equation--6th Powers
• Eric W. Weisstein, Waring's Problem

Crossrefs

Cf. A161882, A161883, A161884, A161885.

Programs

Maple

M:= 10^8:
R:= Vector(M, 74, datatype=integer[4]):
for p from 1 to floor(M^(1/6)) do
  p6:= p^6;
  if p > 1 then A[p]:= R[p6] fi;
  R[p6]:= 1;
  for j from p6+1 to M do
    R[j]:= min(R[j], 1+R[j - p6]);
  od
od:
F:= proc(n, k, ub)
   local lb, m, bestyet, res;
   if ub <= 0 then return -1 fi;
   if n <= M then
     if n = 0 then return 0
     elif R[n] > ub then return -1
     else return R[n]
     fi
   fi;
   lb:= floor(n/k^6);
   if lb > ub then return -1 fi;
   bestyet:= ub;
   for m from lb to 0 by -1 do
     res:= procname(n-m*k^6, k-1, bestyet-m);
     if res >= 0 then
       bestyet:= res+m;
     fi
   od:
   return bestyet
end proc:
for n from floor(M^(1/6))+1 to 50 do
   A[n]:= F(n^6, n-1, 73)
od:
seq(A[n], n=2..50); # Robert Israel, Aug 17 2015

Mathematica

a[n_] := Module[{k}, For[k = 7, True, k++, If[IntegerPartitions[n^6, {k}, Range[n-1]^6] != {}, Print[n, " ", k]; Return[k]]]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Jul 29 2023 *)

PARI

a(n,verbose=0,m=6)={N=n^m;for(k=3,64,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&&!(verbose&&print1("/*"n" "v"*/"))&&return(k),1))}

Python

from itertools import count
from sympy.solvers.diophantine.diophantine import power_representation
def A252486(n):
    m = n**6
    for k in count(2):
        try:
            next(power_representation(m,6,k))
        except:
            continue
        return k # Chai Wah Wu, Jun 25 2024

Extensions

More terms from Manfred Scheucher, Aug 15 2015

a(53)-a(66) from Giovanni Resta, Aug 17 2015