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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
Manfred Scheucher, Sage Script
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
MATHEMATICA
a[n_] := Module[{k}, For[k = 7, True, k++, If[IntegerPartitions[n^6, {k}, Range[n-1]^6] != {}, Print[n, " ", k]; Return[k]]]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Jul 29 2023 *)
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
Sequence in context: A033384 A073327 A169639 * A188828 A331224 A123994
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 March 2 09:28 EST 2024. Contains 370461 sequences. (Running on oeis4.)