A129082 a(n) = numerator 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.

1, 3, 11, 25, 123, 53, 275, 581, 5898, 6337, 81839, 52193, 794409, 929481, 611743, 1609819, 24076913, 6686545, 176364550, 32690593, 9049485, 10684919, 281305624, 439838742, 20192641459, 17176118816, 107883019372, 142161870055, 2874353551691, 3214687921599

Offset

1,2

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Example

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.

Maple

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: A129082 := proc(n) numer(r(n)) ; end: for n from 1 do printf("%d, \n", A129082(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-> numer(r(sh(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];
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_] := Numerator[ r[ sh[ Sort[ ContinuedFraction[ HarmonicNumber[n]]]]]];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 20 2017, after Alois P. Heinz *)

Prog

(Magma) Q:=Rationals(); [ Numerator(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

Crossrefs

Cf. A129083, A129084, A129085.

Sequence in context: A184634 A352013 A164303 * A190476 A060746 A111935

Adjacent sequences: A129079 A129080 A129081 * A129083 A129084 A129085

Keyword

frac,nonn

Author

Leroy Quet, Mar 28 2007

Extensions

6 more terms from R. J. Mathar, Jul 30 2009

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

Status

approved