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A252487 Smallest k such that n^7 = a_1^7 + ... + a_k^7 and all a_i are positive integers less than n.

%I A252487 #37 Feb 16 2025 08:33:24
%S A252487 128,28,66,39,28,26,21,20,18,22,22,22,20,21,14,17,14,14,17,16,17,14,
%T A252487 16,13,15,13,12,15,13,15,13,14,13,14,13,13,14,12,12,12,13,12,12,12,11,
%U A252487 13,13,12,12,13,12,12,11,12,11,11,12,12,11,12,9,12,11,11,11
%N A252487 Smallest k such that n^7 = a_1^7 + ... + a_k^7 and all a_i are positive integers less than n.
%C A252487 Inspired by Fermat's Last Theorem: 2 never occurs in this sequence.
%C A252487 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, ...
%C A252487 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.
%C A252487 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
%C A252487 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
%H A252487 Giovanni Resta, <a href="/A252487/b252487.txt">Table of n, a(n) for n = 2..200</a>
%H A252487 Jean-Charles Meyrignac, <a href="http://euler.free.fr/">Computing Minimal Equal Sums Of Like Powers</a>
%H A252487 Manfred Scheucher, <a href="/A252487/a252487_1.sage.txt">Sage Script for IP-generation</a>
%H A252487 Manfred Scheucher, <a href="/A252487/a252487_2.sage.txt">Sage Script for b-file generation</a>
%H A252487 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiophantineEquation7thPowers.html">Diophantine Equation--7th Powers</a>
%H A252487 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WaringsProblem.html">Waring's Problem</a>
%p A252487 M:= 10^8:
%p A252487 R:= Vector(M,144, datatype=integer[4]):
%p A252487 for p from 1 to floor(M^(1/7)) do
%p A252487   p7:= p^7;
%p A252487   if p > 1 then A[p]:= R[p7] fi;
%p A252487   R[p7]:= 1;
%p A252487   for j from p7+1 to M do
%p A252487     R[j]:= min(R[j],1+R[j - p7]);
%p A252487   od
%p A252487 od:
%p A252487 F:= proc(n,k,ub)
%p A252487    local lb, m, bestyet, res;
%p A252487    if ub <= 0 then return -1 fi;
%p A252487    if n <= M then
%p A252487      if n = 0 then return 0
%p A252487      elif R[n] > ub then return -1
%p A252487      else return R[n]
%p A252487      fi
%p A252487    fi;
%p A252487    lb:= floor(n/k^7);
%p A252487    if lb > ub then return -1 fi;
%p A252487    bestyet:= ub;
%p A252487    for m from lb to 0 by -1 do
%p A252487      res:= procname(n-m*k^7, k-1, bestyet-m);
%p A252487      if res >= 0 then
%p A252487        bestyet:= res+m;
%p A252487      fi
%p A252487    od:
%p A252487    return bestyet
%p A252487 end proc:
%p A252487 for n from floor(M^(1/7))+1 to 50 do
%p A252487    A[n]:= F(n^7,n-1,144)
%p A252487 od:
%p A252487 seq(A[n],n=2..50); # _Robert Israel_, Aug 17 2015
%o A252487 (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))}
%Y A252487 Cf. A002804, A161882, A161883, A161884, A161885, A252486.
%K A252487 nonn
%O A252487 2,1
%A A252487 _M. F. Hasler_, Dec 17 2014
%E A252487 More terms from _Manfred Scheucher_, Aug 15 2015
%E A252487 a(50)-a(66) from _Giovanni Resta_, Aug 17 2015