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Number of distinct unit fractions required to sum to n when using the "splitting algorithm".
1

%I #12 Jul 03 2017 02:03:46

%S 1,4,16,172,4331,232388,4865293065,40149851165417480,

%T 18146043304242768613568943751063,

%U 5522398183372890742378015411585945396419106762128927

%N Number of distinct unit fractions required to sum to n when using the "splitting algorithm".

%C The splitting algorithm decomposes a rational p/q to distinct unit fractions by first creating the multiset with p copies of 1/q, then repeatedly replacing a duplicated element 1/q' with the pair 1/(q'+1), 1/q'(q'+1) until no duplicates remain.

%H Hugo van der Sanden and others, <a href="/A130691/b130691.txt">Table of n, a(n) for n = 1..14</a>

%H L. Beeckmans, <a href="https://doi.org/10.1006/jnth.1993.1015">The Splitting Algorithm for Egyptian Fractions</a>, J. Number Th. 43, 173-185, 1993.

%H Hugo van der Sanden and others, <a href="/A130691/a130691.txt">Table of n, a(n) for n = 1..17</a> [Included as an "a-file", since the last three terms exceed the limit for terms in b-files.]

%e For n=2, the algorithm generates the multisets {1/1, 1/1}, {1/1, 1/2, 1/2}, {1/1, 1/2, 1/3, 1/6}. The final multiset has no duplicate elements, so the algorithm terminates, and has 4 elements, so a(2)=4.

%Y Cf. A002966. - _Robert G. Wilson v_, Jun 10 2010

%K nonn,nice

%O 1,2

%A _Hugo van der Sanden_, Jun 10 2010, with contributions from _Franklin T. Adams-Watters_ and _Robert Gerbicz_