A363937 Minimal number of terms of an Egyptian fraction to be added to, or subtracted from, harmonic number H(n) to get an integer.

0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 2, 3, 4, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 4, 5, 5, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6

Offset

1,5

Comments

The shortest Egyptian fractions for H(n) - floor(H(n)) and ceiling(H(n)) - H(n) are calculated and the smaller length of those fractions is a(n).

Links

Table of n, a(n) for n=1..45.

Ron Knott, Egyptian Fractions

Example

For n = 2: H(2) = 3/2 which is between 1 and 2 and they are reached by the same H(2) - 1 = 2 - H(2) = 1/2 which is 1 term, so a(2) = 1.
For n = 5: H(5) = 137/60 is between 2 and 3; going up 3 - H(5) = 1/2 + 1/6 + 1/20 is 3 terms but going down H(5) - 2 = 1/5 + 1/12 is 2 terms, so the latter is shorter and a(5) = 2 terms.

Mathematica

(* Thanks to Ron Knott for the algorithm. Slow for n>15. *)
check[f_, k_]:= (If[Numerator@f == 1, Return@True];
  If[k == 1, Return@False];
  Catch[Do[
    If[check[f - 1/i, k - 1], Throw@True],
    {i, Range[Ceiling[1/f], Floor[k/f]]}];
   Throw@False]
  );
a[n_]:= (h = HarmonicNumber[n];
  d = {h - Floor[h], Ceiling[h] - h};
  j = 1;
  While[Not[Or @@ (check[#, j] & /@ d)], j++]; j);

Crossrefs

Cf. A281530

Sequence in context: A348459 A266123 A115230 * A338686 A338687 A331677

Adjacent sequences: A363934 A363935 A363936 * A363938 A363939 A363940

Keyword

nonn,more

Author

Denis Ivanov, Jun 29 2023

Extensions

a(31)-a(45) from Dmitry Petukhov, Jul 24 2023

Status

approved