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A252486 Smallest k such that n^6 = a_1^6+...+a_k^6 where all the a_i are positive integers less than n. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

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

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

PROG

(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))}

CROSSREFS

Cf. A161882, A161883, A161884, A161885.

Sequence in context: A033384 A073327 A169639 * A188828 A331224 A123994

Adjacent sequences:  A252483 A252484 A252485 * A252487 A252488 A252489

KEYWORD

nonn

AUTHOR

M. F. Hasler, Dec 17 2014

EXTENSIONS

More terms from Manfred Scheucher, Aug 15 2015

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

STATUS

approved

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Last modified February 25 08:48 EST 2020. Contains 332221 sequences. (Running on oeis4.)