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A192881 Number of terms for the shortest Egyptian fraction representation of 1 starting with 1/n.

%I A192881 #43 Jan 15 2018 23:31:52
%S A192881 1,3,5,8,10,11,13,15,17,19
%N A192881 Number of terms for the shortest Egyptian fraction representation of 1 starting with 1/n.
%C A192881 An Egyptian fraction representation of a rational number a/b is a list of distinct unit fractions with sum a/b.
%H A192881 Mohammad K. Azarian, <a href="http://www.jstor.org/stable/10.4169/college.math.j.42.4.329">Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Problem 958</a>, College Mathematics Journal, Vol. 42, No. 4, September 2011, p. 330.
%H A192881 Mohammad K. Azarian, <a href="http://www.jstor.org/stable/10.4169/college.math.j.43.4.337">Sylvester's Sequence and the Infinite Egyptian Fraction Decomposition of 1, Solution</a> College Mathematics Journal, Vol. 43, No. 4, September 2012, pp. 340-342.
%H A192881 M. N. Bleicher, <a href="http://dx.doi.org/10.1016/0022-314X(72)90069-8">A new algorithm for the expansion of Egyptian fractions</a>, J. Numb. Theory 4 (1972) 342-382
%H A192881 Javier Múgica, <a href="/A192881/a192881.txt">decompositions</a> achieving the terms in this sequence.
%H A192881 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%F A192881 a(n) >= A103762(n) - n + 1.
%e A192881 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_
%Y A192881 Cf. A103762, A294651.
%K A192881 nonn,more,hard
%O A192881 1,2
%A A192881 _Teena Carroll_, Jul 11 2011
%E A192881 Two more terms from _Javier Múgica_, Dec 18 2017