|
%I M2981 #91 Jul 30 2024 14:41:47
%S 1,1,3,14,147,3462,294314,159330691
%N Egyptian fractions: number of solutions of 1 = 1/x_1 + ... + 1/x_n where 0 < x_1 <= ... <= x_n.
%C All denominators in the expansion 1 = 1/x_1 + ... + 1/x_n are bounded by A000058(n-1), i.e., 0 < x_1 <= ... <= x_n < A000058(n-1). Furthermore, for a fixed n, x_i <= (n+1-i)*(A000058(i-1)-1). - _Max Alekseyev_, Oct 11 2012
%C From _R. J. Mathar_, May 06 2010: (Start)
%C This is the leading edge of the triangle A156869. This is also the row n=1 of an array T(n,m) which gives the number of ways to write 1/n as a sum over m (not necessarily distinct) unit fractions:
%C 1, 1, 3, 14, 147, 3462, 294314, ...
%C 1, 2, 10, 108, 2892, 270332, ...
%C 1, 2, 21, 339, 17253, ...
%C 1, 3, 28, 694, 51323, ...
%C ...
%C T(.,2) = A018892. T(.,3) = A004194. T(.,4) = A020327, T(.,5) = A020328. T(2,6) is computed by _D. S. McNeil_, who conjectures that the 2nd row is A003167. (End)
%C If on the other hand, all x_k must be unique, see A006585. - _Robert G. Wilson v_, Jul 17 2013
%D R. K. Guy, Unsolved Problems in Number Theory, D11.
%D D. Singmaster, The number of representations of one as a sum of unit fractions, unpublished manuscript, 1972.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Matthew Brendan Crawford, <a href="https://hdl.handle.net/10919/90573">On the Number of Representations of One as the Sum of Unit Fractions</a>, Master's Thesis, Virginia Polytechnic Institute and State University (2019).
%H Yuya Dan, <a href="http://www.m-hikari.com/imf-2011/1-4-2011/danIMF1-4-2011.pdf">Representation of one as the sum of unit fractions</a>, International Mathematical Forum 6:1 (2011), pp. 25-30.
%H Jacques Le Normand, <a href="http://www.magnesium.net/~palomer/egypt.cpp">C++ code for a(8)</a> [Broken link]
%H Jacques Le Normand, <a href="/A002966/a002966.txt">C++ code for a(8)</a> [Cached copy]
%H Joel Louwsma, <a href="https://arxiv.org/abs/2402.09515">On solutions of Sum_{i=1..n} 1/x_i = 1 in integers of the form 2^a*k^b, where k is a fixed odd positive integer</a>, arXiv:2402.09515 [math.NT], 2024.
%H David Singmaster, <a href="/A002966/a002966.pdf">The number of representations of one as a sum of unit fractions</a>, Unpublished M.S., 1972
%H R. G. Wilson, v, <a href="/A002966/a002966_1.pdf">Fax to N. J. A. Sloane, Sep 9, 1994, with copy of Scientific American column by Ian Stewart</a>
%H <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%F a(n) <= binomial(A007018(n), n-1). - _Charles R Greathouse IV_, Jul 29 2024
%e For n=3 the 3 solutions are {2,3,6}, {2,4,4}, {3,3,3}.
%e For n=4 the solutions are: {2,3,7,42}, {2,3,8,24}, {2,3,9,18}, {2,3,10,15}, {2,3,12,12}, {2,4,5,20}, {2,4,6,12}, {2,4,8,8}, {2,5,5,10}, {2,6,6,6}, {3,3,4,12}, {3,3,6,6}, {3,4,4,6}, {4,4,4,4}. [Neven Juric, May 14 2008]
%o (PARI) a(n,rem=1,mn=1)=if(n==1,return(numerator(rem)==1)); sum(k=max(1\rem+1,mn), n\rem, a(n-1,rem-1/k,k)) \\ _Charles R Greathouse IV_, Jan 04 2015
%Y Cf. A002967, A006585, A000058, A348625.
%K nonn,nice,hard,more
%O 1,3
%A _N. J. A. Sloane_, _Robert G. Wilson v_
%E a(7) from _Jud McCranie_, Nov 15 1999. Confirmed by _Marc Paulhus_.
%E a(8) from John Dethridge (jcd(AT)ms.unimelb.edu.au) and Jacques Le Normand (jacqueslen(AT)sympatico.ca), Jan 06 2004
|
