A097048 a(n) = least denominator Y of the proper fractions X/Y which need n or more terms as an Egyptian fraction.

2, 3, 5, 11, 17, 79, 733, 27539

Offset: 1

Author

Ed Pegg Jr and Don Reble, Jul 21 2004

Keywords

• nonn
• more
• frac
• nice

Comments

These are the simplest proper fractions requiring n parts as an Egyptian fraction, where "simplest" means smallest denominator and the smallest numerator breaks ties: 1/2, 2/3, 4/5, 8/11, 16/17, 77/79, 732/733, ...

Checking just (p-1)/p for prime p finds no example requiring 9 parts for p <= 800399: see "results-single" in the github link. - Hugo van der Sanden, Feb 28 2015

Examples

			27538/27539 is the simplest rational that cannot be expressed as the sum of 7 or fewer distinct unit fractions. That is, no rational p/q requires 8 or more with 0 < p/q < 1, and either q < 27539 or (q = 27539 and p < 27538). - _Hugo van der Sanden_, Sep 14 2010
		

References

• R. K. Guy, Unsolved Problems in Number Theory, D11

Links

• David Eppstein, Ten Algorithms for Egyptian Fractions
• Hugo van der Sanden, code and results on github.

Crossrefs

See A097049 for numerators.

Extensions

a(8) from Hugo van der Sanden, Sep 14 2010