A252476 -id:A252476 - cp's OEIS frontend

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

M. F. Hasler, Dec 17 2014

nonn

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

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

Cf. A161882, A161883, A161884, A161885.

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

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 *)

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))}

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

More terms from Manfred Scheucher, Aug 15 2015

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

A261572 Minimum k such that k^6 can be expressed as the sum of n positive 6th powers.

Original entry on oeis.org

1141, 251, 54, 39, 18, 17, 16, 14, 4, 10, 11, 12, 9, 10, 7, 6, 8, 8, 9, 10, 10, 7, 5, 8, 8, 9, 9, 10, 7, 3, 8, 8, 9, 9, 10, 7, 5, 8, 8, 9, 9, 10, 7, 4, 8, 8, 9, 9, 10, 7, 4, 8, 8, 9, 9, 10, 7, 2, 8, 8, 9, 9, 10, 7, 5, 8, 8, 9, 9, 10, 7, 4, 8, 8, 9, 9, 10, 7, 4

Offset: 7

Jon E. Schoenfield, Aug 24 2015

nonn

It is not known whether there exists a 6th power that can be expressed as the sum of 6 positive 6th powers.

			a(7) = 1141 because 1141^6 = 1077^6 + 894^6 + 702^6 + 474^6 + 402^6 + 234^6 + 74^6 and no integer smaller than 1141 can be expressed as the sum of 7 positive 6th powers.

Eric W. Weisstein, Diophantine Equation--6th Powers

Cf. A252476, A252486.

cp's OEIS Frontend

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

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