A129084 - OEIS

%I #19 Mar 21 2017 04:13:16

%S 1,3,7,25,88,49,219,416,4896,4523,68559,40460,613441,791549,487091,

%T 1123701,16678867,4363873,121113412,24252821,5893113,7436454,

%U 217867766,306700798,14495108003,11420114688,78503059517,93975842393

%N 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.

%H Alois P. Heinz, <a href="/A129084/b129084.txt">Table of n, a(n) for n = 1..750</a>

%e 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.

%p with(numtheory):

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

%p r:= proc(l) local j;

%p       infinity; for j from nops(l) to 1 by -1 do l[j]+1/% od

%p     end:

%p 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:

%p a:= n-> numer(r(hs(sort(cfrac(H(n), 'quotients'))))):

%p seq(a(n), n=1..40);  # _Alois P. Heinz_, Aug 04 2009

%t r[l_] := Module[{lj, j}, For[lj = Infinity; j = Length[l], j >= 1, j--, lj = l[[j]] + 1/lj]; lj];

%t 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];

%t a[n_] := Numerator[ r[ hs[ Sort[ ContinuedFraction[ HarmonicNumber[n]]]]]];

%t Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Mar 20 2017, after _Alois P. Heinz_ *)

%Y Cf. A129082, A129083, A129085.

%K frac,nonn

%O 1,2

%A _Leroy Quet_, Mar 28 2007

%E More terms from _Diana L. Mecum_, Jun 16 2007

%E Extended beyond a(12) _Alois P. Heinz_, Aug 04 2009