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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.
4
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
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
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