1,3

Leroy Quet, Home Page (listed in lieu of email address)

The continued fraction for H(5) = 137/60 is [2;3,1,1,8]. The maximum value a continued fraction can have with these same terms in some order is [8;1,3,1,2] = 123/14.

H := proc(n) add(1/k, k=1..n) ; end: Ltoc := proc(L) numtheory[nthconver](L, nops(L)-1) ; end: r := proc(n) option remember ; local m, rL, rp, L ; if n = 1 then 1; else rL := numtheory[cfrac](H(n), 'quotients') ; rp := combinat[permute](rL) ; m := Ltoc(rL) ; for L in rp do m := max(m, Ltoc(L)) ; od: m ; fi; end: A129083 := proc(n) denom(r(n)) ; end: for n from 1 do printf("%d, \n", A129083(n)) ; od: [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 30 2009]

Contribution from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 04 2009: (Start)

with (numtheory): H:= proc(n) option remember; `if` (n=1, 1, H(n-1)+1/n) end:

r:= proc(l) local j; infinity; for j from nops(l) to 1 by -1 do l[j]+1/% od end:

sh:= proc(l) local ll, h, s, m; ll:= []; h:= nops(l); s:= 1; m:= h; while s<=h do ll:= [ll[], l[m]]; if m=h then h:= h-1; m:= s else s:= s+1; m:= h fi od; ll end:

a:= n-> denom (r (sh (sort (cfrac (H(n), 'quotients'))))): seq (a(n), n=1..40); (End)

(MAGMA) Q:=Rationals(); [ Denominator(Max([ r: r in R ])) where R:=[ c[1, 1]/c[2, 1]: c in C ] where C:=[ Convergents(s): s in S ] where S:=Seqset([ [m(p[i]):i in [1..#x] ]: p in P ]) where m:=map< x->y | [<x[i], y[i]>:i in [1..#x] ] > where P:=Permutations(Seqset(x)) where x:=[1..#y]: y in [ ContinuedFraction(h): h in [ &+[ 1/k: k in [1..n] ]: n in [1..8] ] ] ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jul 31 2009]

Cf. A129082, A129084, A129085.

Sequence in context: A057648 A068511 A060590 this_sequence A045685 A045676 A098288

Adjacent sequences: A129080 A129081 A129082 this_sequence A129084 A129085 A129086

frac,nonn

Leroy Quet Mar 28 2007

6 more terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 30 2009

Extended byond a(12) Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 04 2009