A281530 -id:A281530 - cp's OEIS frontend

A281527 Triangle read by rows: T(n,k) = minimal number of terms needed to write k/n as a signed sum of distinct unit fractions, 1 <= k < n.

1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 3, 2, 1, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 3, 3, 2, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 3, 3, 3, 3, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 3, 3, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 3, 2

Offset: 2

Arkadiusz Wesolowski, Jan 23 2017

			The triangle T(n,k) begins:
2:                     1
3:                   1   2
4:                 1   1   2
5:               1   2   2   2
6:             1   1   1   2   2
7:           1   2   2   2   3   2
8:         1   1   2   1   2   2   2
9:       1   2   1   2   2   2   3   2

Index entries for sequences related to Egyptian fractions

Cf. A281530.

A281531 a(n) is the least numerator k such that the proper fraction k/n needs three or more terms as an Egyptian fraction, or 0 if no such numerator exists.

Original entry on oeis.org

0, 0, 0, 4, 0, 3, 7, 7, 8, 5, 11, 3, 6, 7, 7, 4, 13, 3, 13, 9, 5, 5, 17, 4, 6, 8, 12, 4, 14, 3, 7, 5, 8, 11, 17, 3, 6, 9, 17, 4, 18, 3, 7, 11, 7, 5, 21, 3, 8, 7, 11, 4, 13, 9, 13, 7, 7, 7, 28, 3, 5, 13, 7, 4, 10, 3, 11, 11, 13, 5, 23, 3, 6, 11, 9, 5, 11, 3, 19

Offset: 2

Arkadiusz Wesolowski, Jan 23 2017

nonn

If n > 3 is prime, a(n) = A007978(n+1). - Robert Israel, Dec 26 2019

Robert Israel, Table of n, a(n) for n = 2..10000
Index entries for sequences related to Egyptian fractions

Cf. A007978, A281530.

lst:=[]; for n in [2..80] do for k in [1..n-1] do f:=k/n; x:=1; v:=0; if Numerator(f) eq 1 then v:=1; else while f lt 2/x do if Numerator(f-1/x) eq 1 then v:=1; break; end if; x+:=1; end while; end if; if v eq 0 then Append(~lst, k); break; end if; if k eq n-1 then Append(~lst, 0); end if; end for; end for; lst;

f:= proc(n) option remember; local k,T;
  T:= numtheory:-divisors(n^2);
  for k from 2 to n-1 do
    g:= igcd(k,n);
    if g > 1 then
       r:= procname(n/g);
       if k = r*g then return k fi;
    else
       if not member(-n mod k,  T mod k) then return k fi
    fi
  od;
0
end proc;
map(f, [$2..100]); # Robert Israel, Dec 25 2019

a[n_] := a[n] = Module[{k, T}, T = Divisors[n^2]; For[k = 2, k <= n - 1, k++, g = GCD[k, n]; If[g > 1, r = a[n/g]; If[k == r g, Return [k]], If[FreeQ[Mod[T, k], Mod[-n, k]], Return [k]]]]; 0];
a /@ Range[2, 100] (* Jean-François Alcover, Oct 06 2020, after Robert Israel *)

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

Original entry on oeis.org

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

Denis Ivanov, Jun 29 2023

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).

			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.

Ron Knott, Egyptian Fractions

Cf. A281530

(* 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);

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

cp's OEIS Frontend

A281527 Triangle read by rows: T(n,k) = minimal number of terms needed to write k/n as a signed sum of distinct unit fractions, 1 <= k < n.

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A281531 a(n) is the least numerator k such that the proper fraction k/n needs three or more terms as an Egyptian fraction, or 0 if no such numerator exists.

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

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Programs

Mathematica

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