A050205 Triangle read by rows: number of terms in unit fraction representation of k/n using the greedy algorithm, 1<=k<=n-1.

1, 1, 2, 1, 1, 2, 1, 2, 2, 3, 1, 1, 1, 2, 2, 1, 2, 3, 2, 3, 3, 1, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 3, 3, 1, 1, 2, 2, 1, 2, 2, 3, 3, 1, 2, 2, 2, 3, 2, 3, 4, 4, 4, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 3, 3, 3, 4, 1, 2, 1, 2, 1, 2, 3, 2, 2, 2, 3

Offset

2,3

Links

Seiichi Manyama, Rows n = 2..141, flattened

Eric Weisstein's World of Mathematics, Unit Fraction.

Wikipedia, Greedy algorithm for Egyptian fractions.

Formula

T(n,k) >= A097847(n,k). - Pontus von Brömssen, May 11 2026

Example

n\k | 1  2  3  4  5  6  7  8
----*------------------------
  2 | 1;
  3 | 1, 2;
  4 | 1, 1, 2;
  5 | 1, 2, 2, 3;
  6 | 1, 1, 1, 2, 2;
  7 | 1, 2, 3, 2, 3, 3;
  8 | 1, 1, 2, 1, 2, 2, 3;
  9 | 1, 2, 1, 2, 2, 2, 3, 3;
  ...
T(3,2) = 2 because the greedy representation of 2/3 has 2 terms: 2/3 = 1/2 + 1/6.
T(17,4) = 4 because the greedy representation of 4/17 has 4 terms: 4/17 = 1/5 + 1/29 + 1/1233 + 1/3039345. This is the first case where the greedy representation does not have the minimum possible number of terms (A097847(17,4) = 3).

Crossrefs

Cf. A050206, A050210 (Largest denominator), A097847, A100678 (main diagonal), A260618.

Sequence in context: A025836 A029319 A243987 * A281530 A340260 A175190

Adjacent sequences: A050202 A050203 A050204 * A050206 A050207 A050208

Keyword

nonn,easy,tabl

Author

Eric W. Weisstein

Extensions

Offset changed to 2 by Seiichi Manyama, Sep 18 2022

Status

approved