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A129083
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a(n) = denominator of b(n): b(n) = the maximum possible value for a continued fraction whose terms are a permutation of the terms of the simple continued fraction for H(n) = sum{k=1 to n} 1/k, the n-th harmonic number.
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4
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1, 1, 2, 2, 14, 12, 47, 74, 751, 455, 1609, 3281, 115912, 23276, 13920, 162643, 1613467, 107881, 6816743, 1825459, 245428, 769416, 48264683, 24515971, 1069940948, 821503527, 1565820591, 3095313533, 110858310133, 16156751822
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OFFSET
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1,3
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LINKS
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EXAMPLE
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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.
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MAPLE
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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: # R. J. Mathar, Jul 30 2009
# second Maple program:
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'))))):
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MATHEMATICA
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r[l_] := Module[{lj, j}, For[lj = Infinity; j = Length[l], j >= 1, j--, lj = l[[j]] + 1/lj]; lj];
sh[l_] := Module[{ll, h, s, m}, ll = {}; h = Length[l]; s = 1; m = h; While[s <= h, ll = Append[ll, l[[m]]]; If[m == h, h--; m = s, s++; m = h ]]; ll];
a[n_] := Denominator[ r[ sh[ Sort[ ContinuedFraction[ HarmonicNumber[n]]]]] ];
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PROG
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(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] ] ] ]; // Klaus Brockhaus, Jul 31 2009
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CROSSREFS
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KEYWORD
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frac,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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