A002966 Egyptian fractions: number of solutions of 1 = 1/x_1 + ... + 1/x_n where 0 < x_1 <= ... <= x_n. (Formerly M2981)

1, 1, 3, 14, 147, 3462, 294314, 159330691

Offset

1,3

Comments

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

From R. J. Mathar, May 06 2010: (Start)

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:

1, 1, 3, 14, 147, 3462, 294314, ...

1, 2, 10, 108, 2892, 270332, ...

1, 2, 21, 339, 17253, ...

1, 3, 28, 694, 51323, ...

...

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)

If on the other hand, all x_k must be unique, see A006585. - Robert G. Wilson v, Jul 17 2013

References

R. K. Guy, Unsolved Problems in Number Theory, D11.

D. Singmaster, The number of representations of one as a sum of unit fractions, unpublished manuscript, 1972.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..8.

Matthew Brendan Crawford, On the Number of Representations of One as the Sum of Unit Fractions, Master's Thesis, Virginia Polytechnic Institute and State University (2019).

Yuya Dan, Representation of one as the sum of unit fractions, International Mathematical Forum 6:1 (2011), pp. 25-30.

Jacques Le Normand, C++ code for a(8) [Broken link]

Jacques Le Normand, C++ code for a(8) [Cached copy]

Joel Louwsma, 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, arXiv:2402.09515 [math.NT], 2024.

David Singmaster, The number of representations of one as a sum of unit fractions, Unpublished M.S., 1972

R. G. Wilson, v, Fax to N. J. A. Sloane, Sep 9, 1994, with copy of Scientific American column by Ian Stewart

Index entries for sequences related to Egyptian fractions

Formula

a(n) <= binomial(A007018(n), n-1). - Charles R Greathouse IV, Jul 29 2024

Example

For n=3 the 3 solutions are {2,3,6}, {2,4,4}, {3,3,3}.
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]

Prog

(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

Crossrefs

Cf. A002967, A006585, A000058, A348625.

Sequence in context: A073550 A319361 A367422 * A075654 A330603 A261006

Adjacent sequences: A002963 A002964 A002965 * A002967 A002968 A002969

Keyword

nonn,nice,hard,more

Author

N. J. A. Sloane, Robert G. Wilson v

Extensions

a(7) from Jud McCranie, Nov 15 1999. Confirmed by Marc Paulhus.

a(8) from John Dethridge (jcd(AT)ms.unimelb.edu.au) and Jacques Le Normand (jacqueslen(AT)sympatico.ca), Jan 06 2004

Status

approved