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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. 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 (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, ... 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 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 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 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 for IP-generation Manfred Scheucher, Sage Script for b-file generation Eric Weisstein's World of Mathematics, Diophantine Equation--7th Powers Eric Weisstein's World of Mathematics, Waring's Problem 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(n-m*k^7, k-1, bestyet-m); 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, n-1, 144) od: seq(A[n], n=2..50); # Robert Israel, Aug 17 2015 PROG (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))} CROSSREFS Cf. A002804, A161882, A161883, A161884, A161885, A252486. Sequence in context: A303324 A121374 A336774 * A160638 A188829 A336776 Adjacent sequences: A252484 A252485 A252486 * A252488 A252489 A252490 KEYWORD nonn AUTHOR M. F. Hasler, Dec 17 2014 EXTENSIONS More terms from Manfred Scheucher, Aug 15 2015 a(50)-a(66) from Giovanni Resta, Aug 17 2015 STATUS approved

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Last modified April 14 17:14 EDT 2024. Contains 371666 sequences. (Running on oeis4.)