A130738 Greedy odd Egyptian fraction representation of 1 (without repeats).

3, 5, 7, 9, 11, 13, 23, 721, 979007, 661211444787, 622321538786143185105739, 511768271877666618502328764212401495966764795565, 209525411280522638000804396401925664136495425904830384693383280180439963265695525939102230139815

Offset: 1

Author

Jon Wild, Jul 06 2007

Keywords

• nonn
• fini
• full

Comments

a(n) is the largest odd Egyptian fraction as yet unused, such that the sum of the Egyptian fractions so far does not exceed 1. The sum of a(n) is a greedy representation (greedy because each step bites off as much as possible) of 1, using only odd Egyptian fractions, all distinct.

Terms a(11)-a(13) were found by David Eppstein (see posting from Nov 09 1996), who says that he found them by applying EgyptOddGreedy[2/3,5] from his Egyptian fractions notebook.

Examples

			E.g. a(8)=721 because 1/721 is the largest odd Egyptian fraction less than 1-1/a(1)-1/a(2)-1/a(3)-1/a(4)-1/a(5)-1/a(6)-1/a(7).
1/3 + 1/5 + 1/7 + 1/9 + 1/11 + 1/13 + 1/23 + 1/721 + 1/979007 + 1/661211444787 + 1/622321538786143185105739 + 1/511768271877666618502328764212401495966764795565 + 1/209525411280522638000804396401925664136495425904830384693383280180439963265695525939102230139815 = 1.
		

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

Links

• David Eppstein, Egyptian fractions
• David Eppstein, Egyptian fractions, Discussion, Nov 09 1996.
• Index entries for sequences related to Egyptian fractions

Crossrefs

Cf. A002966, A169820.

Extensions

Edited and a(11)-a(13) added by N. J. A. Sloane, May 29 2010, at the suggestion of Jan Szejko.