

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
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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 bfile 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 IPsolver. 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



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(nm*k^7, k1, bestyetm);
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, n1, 144)
od:


PROG

(PARI) a(n, verbose=0, m=7)={N=n^m; for(k=3, 999, forvec(v=vector(k1, i, [1, n\sqrtn(k+1i, m)]), ispower(Nsum(i=1, k1, v[i]^m), m, &K)&&K>0&&!(verbose&&print1("/*"n" "v"*/"))&&return(k), 1))}


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



