A216975 Triangle read by rows in which row n gives the lexicographically earliest minimal sum denominators among all possible n-term Egyptian fractions with unit sum.

1, 0, 0, 2, 3, 6, 2, 4, 6, 12, 3, 4, 5, 6, 20, 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, 9, 10, 11, 12, 14, 15, 18, 22, 28, 33, 7, 8, 9, 10, 11, 12, 14, 15, 18, 22, 24, 28, 33

Offset

1,4

Comments

This sequence is the lexicographically earliest Egyptian fraction (denominators only) describing the minimal sum given in A213062.

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

References

Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330. Solution published in Vol. 43, No. 4, September 2012, pp. 340-342

Links

Robert Price, Rows n = 1..24, flattened

Harry Ruderman and Paul Erdős, Problem E2427: Bounds for Egyptian fraction partitions of unity (comments), Amer. Math. Monthly, 1974 (Vol. 81), pp. 780-782.

Eric Weisstein's World of Mathematics, Egyptian Fraction

Wikipedia, Egyptian fraction

Index entries for sequences related to Egyptian fractions

Example

Row 5 = [3,4,5,6,20]: lexicographically earliest minimal sum (38) denominators among 72 possible 5-term Egyptian fractions with unit sum.
1 = 1/3 + 1/4 + 1/5 + 1/6 + 1/20.
Triangle begins:
1;
0, 0;
2, 3, 6;
2, 4, 6, 12;
3, 4, 5,  6, 20;
3, 4, 6, 10, 12, 15;

Crossrefs

Cf. A030659, A073546, A213062, A216993.

Sequence in context: A275734 A216993 A073546 * A275666 A330666 A319432

Adjacent sequences: A216972 A216973 A216974 * A216976 A216977 A216978

Keyword

nonn,tabl

Author

Robert Price, Sep 21 2012

Status

approved