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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 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 Eric W. Weisstein, Diophantine Equation--6th Powers 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 Cf. A161882, A161883, A161884, A161885. Sequence in context: A033384 A073327 A169639 * A188828 A331224 A123994 Adjacent sequences: A252483 A252484 A252485 * A252487 A252488 A252489 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.)