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%I #18 Dec 12 2015 16:40:20
%S 12,84,3612,6526884,21300113901612
%N Largest denominators in even Egyptian fraction representation of 1.
%C An Egyptian Fraction representation of a rational number a/b is a list of distinct unit fractions with sum a/b. We will call it an even Egyptian Fraction representation if only even integers are used as denominators. The n-th term of this sequence gives the largest denominators that appear in an Egyptian Fraction Representation of one with length n+3. For instance, 1/2 + 1/4 + 1/6 + 1/12 gives a 4-term even representation of one, which is the shortest possible Egyptian even fraction representation of one.
%C This sequence can be derived from the Sylvester sequence (A000058). If s(n) represents the Sylvester sequence, s(n)-1 is the largest denominator appearing in an n term Egyptian fraction representation of 1. There is a one-to-one correspondence between k-term representations and (k+1)-term even representation for k<12. An even representation has to have at least 4 terms, thus a(1) is related to s(3). a(1) = 2*(s(3) - 1), etc.
%C Excluding the first two terms of the Pythagorean spiral sequence (A053631) yields this sequence.
%D 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. Solution published in Vol. 43, No. 4, September 2012, pp. 340-342
%F a(n+1) = 1/2*a(n)^2 + a(n).
%e a(2) = 1/2*12^2 + 12 so a(2) = 84.
%Y Cf. A000058, A007018, A053631.
%Y A053631 and A127690 are very similar to this sequence.
%K nonn
%O 1,1
%A _Teena Carroll_, Jul 06 2011
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