A192881 Number of terms for the shortest Egyptian fraction representation of 1 starting with 1/n.
1, 3, 5, 8, 10, 11, 13, 15, 17, 19
Offset: 1
Examples
Since 1/3 + 1/4 + 1/5 + 1/6 + 1/20 = 1, we see that a(3) <= 5. We know the maximum sum of 4 distinct unit fractions (1/3 or less) is 19/20, so this shows a(3)=5. An Egyptian fraction decomposition of 1 starting with 1/4 must have at least 8 terms; however, the expressions need not be unique, as all three of 1 = 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/230 + 1/57960, 1 = 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/231 + 1/27720 and 1 = 1/4 + 1/5 + 1/6 + 1/9 + 1/10 + 1/15 + 1/18 + 1/20 achieve this bound. - _Teena Carroll_, _Haoqi Chen_ and _Javier Múgica_
Links
- 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.
- Mohammad K. Azarian, Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Solution College Mathematics Journal, Vol. 43, No. 4, September 2012, pp. 340-342.
- M. N. Bleicher, A new algorithm for the expansion of Egyptian fractions, J. Numb. Theory 4 (1972) 342-382
- Javier Múgica, decompositions achieving the terms in this sequence.
- Index entries for sequences related to Egyptian fractions
Formula
a(n) >= A103762(n) - n + 1.
Extensions
Two more terms from Javier Múgica, Dec 18 2017
Comments