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

1,2

Comments

An Egyptian fraction representation of a rational number a/b is a list of distinct unit fractions with sum a/b.

Links

Table of n, a(n) for n=1..10.

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.

Example

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

Crossrefs

Cf. A103762, A294651.

Sequence in context: A360030 A120943 A087792 * A189755 A141436 A282897

Adjacent sequences: A192878 A192879 A192880 * A192882 A192883 A192884

Keyword

nonn,more,hard

Author

Teena Carroll, Jul 11 2011

Extensions

Two more terms from Javier Múgica, Dec 18 2017

Status

approved