A129085 a(n) = denominator of b(n): b(n) = the minimum 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.

1, 2, 6, 12, 79, 22, 187, 369, 4343, 4220, 67223, 38067, 535331, 772210, 476254, 1020589, 15631362, 4294584, 116606407, 22970156, 5737508, 6936929, 185961619, 290508289, 13765708850, 10898842249, 77379962122, 91973292918, 1858284737854, 2220029652331

Offset

1,2

Links

Alois P. Heinz, Table of n, a(n) for n = 1..750

Example

The continued fraction for H(5) = 137/60 is [2;3,1,1,8]. The minimum value a continued fraction can have with these same terms in some order is [1;8,1,3,2] = 88/79.

Maple

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:
hs:= proc(l) local ll, h, s, m; ll:= []; h:= nops(l); s:= 1; m:= s; 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(hs(sort(cfrac(H(n), 'quotients'))))):
seq(a(n), n=1..40);  # Alois P. Heinz, Aug 04 2009

Mathematica

r[l_] := Module[{lj, j}, For[lj = Infinity; j = Length[l], j >= 1, j--, lj = l[[j]] + 1/lj]; lj];
hs[l_] := Module[{ll, h, s, m}, ll = {}; h = Length[l]; s = 1; m = s; While[s <= h, ll = Append[ll, l[[m]]]; If[m == h, h--; m = s, s++; m = h ]]; ll];
a[n_] := Denominator[ r[ hs[ Sort[ ContinuedFraction[ HarmonicNumber[n]]]]] ];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 20 2017, after Alois P. Heinz *)

Crossrefs

Cf. A129082, A129083, A129084.

Sequence in context: A166470 A144144 A226178 * A274941 A141288 A038787

Adjacent sequences: A129082 A129083 A129084 * A129086 A129087 A129088

Keyword

frac,nonn

Author

Leroy Quet, Mar 28 2007

Extensions

More terms from Diana L. Mecum, Jun 16 2007

Extended beyond a(12) Alois P. Heinz, Aug 04 2009

Status

approved