A097847 Triangle read by rows: T(n,k) = minimal number of terms needed to write k/n (for 1 <= k <= n) as a sum of unit fractions.

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

Offset

1,5

Links

Table of n, a(n) for n=1..99.

Index entries for sequences related to Egyptian fractions

Formula

T(n, n-1) = A330808(n). - Jon E. Schoenfield, Jan 11 2020

T(n,k) <= A050205(n,k) for 1 <= k <= n-1. - Pontus von Brömssen, May 11 2026

Example

Triangle begins:
  1
  1 1
  1 2 1
  1 1 2 1
  1 2 2 3 1
  1 1 1 2 2 1
  1 2 3 2 3 3 1
  1 1 2 1 2 2 3 1
  1 2 1 2 2 2 3 3 1
  1 1 2 2 1 2 2 3 3 1
  1 2 2 2 3 2 3 4 4 4 1
  ...
T(17,4) = 3 because 4/17 can be written as a sum of 3 (but no fewer) unit fractions, e.g. 4/17 = 1/5 + 1/30 + 1/510. This is the first case where the number of terms is less than the number of terms in the greedy representation (A050205(17,4) = 4).

Crossrefs

Cf. A050205, A097848, A097849, A330808.

Row sums give A270429.

Sequence in context: A174541 A029444 A122191 * A275120 A144379 A337260

Adjacent sequences: A097844 A097845 A097846 * A097848 A097849 A097850

Keyword

nonn,tabl

Author

Franklin T. Adams-Watters, Aug 31 2004

Status

approved