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

1,4

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.

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).

Example

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;
  ...

Crossrefs

Cf. A073546, A216993.

Sequence in context: A275734 A216993 A073546 * A216975 A275666 A330666

Adjacent sequences: A378720 A378721 A378722 * A378729 A378730 A378731

Keyword

nonn,tabl,new

Author

Sean A. Irvine, Dec 05 2024

Status

approved