A129084 a(n) = numerator 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, 3, 7, 25, 88, 49, 219, 416, 4896, 4523, 68559, 40460, 613441, 791549, 487091, 1123701, 16678867, 4363873, 121113412, 24252821, 5893113, 7436454, 217867766, 306700798, 14495108003, 11420114688, 78503059517, 93975842393

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-> numer(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_] := Numerator[ 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, A129085.

Sequence in context: A148737 A148738 A148739 * A287892 A343278 A002870

Adjacent sequences: A129081 A129082 A129083 * A129085 A129086 A129087

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