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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.

%I A252486 #34 Feb 16 2025 08:33:24
%S A252486 64,36,15,29,22,21,15,19,15,17,15,16,14,15,13,12,11,11,13,14,12,13,13,
%T A252486 12,12,12,12,12,11,11,11,11,11,13,11,11,11,10,11,11,11,11,11,10,11,11,
%U A252486 11,11,11,11,11,11,9,11,10,11,11,11,9,10,11,11,11,11,10
%N A252486 Smallest k such that n^6 = a_1^6+...+a_k^6 where all the a_i are positive integers less than n.
%C A252486 Inspired by Fermat's Last Theorem: 2 never occurs in this sequence.
%C A252486 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.
%C A252486 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.
%C A252486 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
%H A252486 Giovanni Resta, <a href="/A252486/b252486.txt">Table of n, a(n) for n = 2..200</a>
%H A252486 Jean-Charles Meyrignac, <a href="http://euler.free.fr/">Computing Minimal Equal Sums Of Like Powers</a>
%H A252486 Manfred Scheucher, <a href="/A252486/a252486.sage.txt">Sage Script</a>
%H A252486 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/DiophantineEquation6thPowers.html">Diophantine Equation--6th Powers</a>
%H A252486 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/WaringsProblem.html">Waring's Problem</a>
%p A252486 M:= 10^8:
%p A252486 R:= Vector(M, 74, datatype=integer[4]):
%p A252486 for p from 1 to floor(M^(1/6)) do
%p A252486   p6:= p^6;
%p A252486   if p > 1 then A[p]:= R[p6] fi;
%p A252486   R[p6]:= 1;
%p A252486   for j from p6+1 to M do
%p A252486     R[j]:= min(R[j], 1+R[j - p6]);
%p A252486   od
%p A252486 od:
%p A252486 F:= proc(n, k, ub)
%p A252486    local lb, m, bestyet, res;
%p A252486    if ub <= 0 then return -1 fi;
%p A252486    if n <= M then
%p A252486      if n = 0 then return 0
%p A252486      elif R[n] > ub then return -1
%p A252486      else return R[n]
%p A252486      fi
%p A252486    fi;
%p A252486    lb:= floor(n/k^6);
%p A252486    if lb > ub then return -1 fi;
%p A252486    bestyet:= ub;
%p A252486    for m from lb to 0 by -1 do
%p A252486      res:= procname(n-m*k^6, k-1, bestyet-m);
%p A252486      if res >= 0 then
%p A252486        bestyet:= res+m;
%p A252486      fi
%p A252486    od:
%p A252486    return bestyet
%p A252486 end proc:
%p A252486 for n from floor(M^(1/6))+1 to 50 do
%p A252486    A[n]:= F(n^6, n-1, 73)
%p A252486 od:
%p A252486 seq(A[n], n=2..50); # _Robert Israel_, Aug 17 2015
%t A252486 a[n_] := Module[{k}, For[k = 7, True, k++, If[IntegerPartitions[n^6, {k}, Range[n-1]^6] != {}, Print[n, " ", k]; Return[k]]]];
%t A252486 Table[a[n], {n, 2, 100}] (* _Jean-François Alcover_, Jul 29 2023 *)
%o A252486 (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))}
%o A252486 (Python)
%o A252486 from itertools import count
%o A252486 from sympy.solvers.diophantine.diophantine import power_representation
%o A252486 def A252486(n):
%o A252486     m = n**6
%o A252486     for k in count(2):
%o A252486         try:
%o A252486             next(power_representation(m,6,k))
%o A252486         except:
%o A252486             continue
%o A252486         return k # _Chai Wah Wu_, Jun 25 2024
%Y A252486 Cf. A161882, A161883, A161884, A161885.
%K A252486 nonn
%O A252486 2,1
%A A252486 _M. F. Hasler_, Dec 17 2014
%E A252486 More terms from _Manfred Scheucher_, Aug 15 2015
%E A252486 a(53)-a(66) from _Giovanni Resta_, Aug 17 2015