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A002966 Egyptian fractions: number of solutions of 1 = 1/x_1 + ... + 1/x_n where 0 < x_1 <= ... <= x_n.
(Formerly M2981)
15
1, 1, 3, 14, 147, 3462, 294314, 159330691 (list; graph; refs; listen; history; text; internal format)
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 counts 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D. Singmaster, ``The number of representations of one as a sum of unit fractions,'' unpublished manuscript, 1972.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.

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

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

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

Index entries for sequences related to Egyptian fractions

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]

CROSSREFS

Cf. A002967, A006585, A000058.

Sequence in context: A096657 A126933 A073550 * A075654 A185238 A090897

Adjacent sequences:  A002963 A002964 A002965 * A002967 A002968 A002969

KEYWORD

nonn,nice,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

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Last modified September 23 01:51 EDT 2014. Contains 247086 sequences.