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

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

Offset: 1

Author

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

Keywords

• nonn
• nice
• hard
• more

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

Examples

			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]
		

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• D. Singmaster, The number of representations of one as a sum of unit fractions, unpublished manuscript, 1972.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

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• David Singmaster, The number of representations of one as a sum of unit fractions, Unpublished M.S., 1972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

Crossrefs

Cf. A002967, A006585, A000058, A348625.

Programs

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

Formula

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

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