login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A252487 Smallest k such that n^7 = a_1^7 + ... + a_k^7 and all a_i are positive integers less than n. 1
128, 28, 66, 39, 28, 26, 21, 20, 18, 22, 22, 22, 20, 21, 14, 17, 14, 14, 17, 16, 17, 14, 16, 13, 15, 13, 12, 15, 13, 15, 13, 14, 13, 14, 13, 13, 14, 12, 12, 12, 13, 12, 12, 12, 11, 13, 13, 12, 12, 13, 12, 12, 11, 12, 11, 11, 12, 12, 11, 12, 9, 12, 11, 11, 11 (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, ... occur first at indices 568, 102, 62, ...

I conjecture that the sequence is bounded by the initial term a(2)=128. Probably even a(4)=66, a(5)=39, a(6)=28 and some more are followed only by smaller terms.

I've uploaded two scripts; one to compute the b-file and one to generate an IP file. For the first script, a parameter kmax can be set to gain a speedup but more memory is used. The other one (which also works with large integers now) should be used in case someone has a good IP-solver. Higher terms might be computable faster with a good IP solver. - Manfred Scheucher, Aug 14 2015

From results on Waring's problem, it is known that all a(n) <= A002804(7) = 143, and a(n) <= 33 for all sufficiently large n. - Robert Israel, Aug 16 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 for IP-generation

Manfred Scheucher, Sage Script for b-file generation

Eric Weisstein's World of Mathematics, Diophantine Equation--7th Powers

Eric Weisstein's World of Mathematics, Waring's Problem

MAPLE

M:= 10^8:

R:= Vector(M, 144, datatype=integer[4]):

for p from 1 to floor(M^(1/7)) do

  p7:= p^7;

  if p > 1 then A[p]:= R[p7] fi;

  R[p7]:= 1;

  for j from p7+1 to M do

    R[j]:= min(R[j], 1+R[j - p7]);

  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^7);

   if lb > ub then return -1 fi;

   bestyet:= ub;

   for m from lb to 0 by -1 do

     res:= procname(n-m*k^7, k-1, bestyet-m);

     if res >= 0 then

       bestyet:= res+m;

     fi

   od:

   return bestyet

end proc:

for n from floor(M^(1/7))+1 to 50 do

   A[n]:= F(n^7, n-1, 144)

od:

seq(A[n], n=2..50); # Robert Israel, Aug 17 2015

PROG

(PARI) a(n, verbose=0, m=7)={N=n^m; for(k=3, 999, 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. A002804, A161882, A161883, A161884, A161885, A252486.

Sequence in context: A035880 A010369 A121374 * A160638 A188829 A172532

Adjacent sequences:  A252484 A252485 A252486 * A252488 A252489 A252490

KEYWORD

nonn

AUTHOR

M. F. Hasler, Dec 17 2014

EXTENSIONS

More terms from Manfred Scheucher, Aug 15 2015

a(50)-a(66) from Giovanni Resta, Aug 17 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified June 28 04:43 EDT 2017. Contains 288813 sequences.