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A192881
Number of terms for the shortest Egyptian fraction representation of 1 starting with 1/n.
4
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
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.
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
Sequence in context: A360030 A120943 A087792 * A189755 A141436 A282897
KEYWORD
nonn,more,hard
AUTHOR
Teena Carroll, Jul 11 2011
EXTENSIONS
Two more terms from Javier Múgica, Dec 18 2017
STATUS
approved