1,2

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

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.

a(n) >= A103762(n) - n + 1.

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

From Max Alekseyev, Oct 30 2025: (Start)

Examples of shortest Egyptian fractions (as lists of denominators):

a(11) = 21: (11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 561, 71820, 1315600)

a(12) = 23: (12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 33, 35, 38, 51, 34884, 1315600)

a(13) = 25: (13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 32, 35, 40, 41, 44, 46, 58449600, 6548068942272)

a(14) = 26: (14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 39, 44, 58, 1384416, 1591995600)

a(15) = 28: (15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 35, 36, 38, 39, 40, 44, 48, 52, 33599412, 130917600)

a(16) = 30: (16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 36, 38, 39, 40, 42, 43, 44, 50, 52, 54, 1735650, 68479214496)

(End)

Cf. A103762, A294651.

Sequence in context: A120943 A360030 A087792 * A189755 A141436 A282897

Adjacent sequences: A192878 A192879 A192880 * A192882 A192883 A192884

nonn,more,hard

Teena Carroll, Jul 11 2011

a(9)-a(10) from Javier Múgica, Dec 18 2017

a(11)-a(16) from Max Alekseyev, Oct 30 2025

approved