A378723 Triangle read by rows: row n gives denominators of n distinct unit fractions (or Egyptian fractions) summing to 1, where denominators are listed in increasing order and the denominators from largest to smallest are as small as possible.

1, 0, 0, 2, 3, 6, 2, 4, 6, 12, 2, 4, 10, 12, 15, 3, 4, 6, 10, 12, 15, 3, 4, 9, 10, 12, 15, 18, 4, 5, 6, 9, 10, 15, 18, 20, 4, 6, 8, 9, 10, 12, 15, 18, 24, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 6, 7, 8, 9, 10, 12, 14, 15, 18, 24, 28, 6, 7, 8, 9, 10, 14, 15, 18, 20, 24, 28, 30

Offset: 1

Author

Sean A. Irvine, Dec 05 2024

Keywords

• nonn
• tabl

Comments

Row 2 = [0,0] corresponds to the fact that 1 cannot be written as an Egyptian fraction with 2 (distinct) terms.

There can be more the one solution with the same smallest maximum denominator. For example, if n=8, we have:

1/3 + 1/5 + 1/9 + 1/10 + 1/12 + 1/15 + 1/18 + 1/20 = 1,

1/4 + 1/5 + 1/6 + 1/9 + 1/10 + 1/15 + 1/18 + 1/20 = 1.

In this sequence, the second solution is taken because 10 < 12 when reading the denominators from the right. In A216993, the first solution is taken because 3 < 4 when reading the denominators from the left.

Examples

			Triangle begins:
  1;
  0, 0;
  2, 3,  6;
  2, 4,  6, 12;
  2, 4, 10, 12, 15;
  3, 4,  6, 10, 12, 15;
  3, 4,  9, 10, 12, 15, 18;
  4, 5,  6,  9, 10, 15, 18, 20;
  4, 6,  8,  9, 10, 12, 15, 18, 24;
  5, 6,  8,  9, 10, 12, 15, 18, 20, 24;
  6, 7,  8,  9, 10, 12, 14, 15, 18, 24, 28;
  6, 7,  8,  9, 10, 14, 15, 18, 20, 24, 28, 30;
  ...
		

References

• R. K. Guy, Unsolved Problems in Number Theory, 2nd Edition, page 161.

Links

• Sean A. Irvine, Table of n, a(n) for n = 1..136
• K. S. Brown, Unit Fractions, smallest last term.
• Sean A. Irvine Java program (github).

Crossrefs

Cf. A073546, A216993.