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A002967 Egyptian fractions: number of solutions of 1 = 1/x_1 + ... + 1/x_n in positive integers.
(Formerly M4745)
1, 1, 10, 215, 12231, 2025462, 1351857641, 6255560531733 (list; graph; refs; listen; history; text; internal format)



Solutions differing only in the order of the x_i are counted as distinct.

All denominators in the expansion 1 = 1/x_1 + ... + 1/x_n are bounded by A000058(n-1)=A129871(n). - Max Alekseyev, Dec 30 2003


Mohammad K. Azarian, Diophantine Pair, Problem B-881, Fibonacci Quarterly, Vol. 37, No. 3, August 1999, pp. 277-278.  Solution published in Vol. 38, No. 2, May 2000, pp. 183-184.

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D. Singmaster, "The number of representations of one as a sum of unit fractions," unpublished manuscript, 1972.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


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

Zachary Harris, Joel Louwsma, On Arithmetical Structures on Complete Graphs, arXiv:1909.02022 [math.NT], 2019.

Putnam Competition, 58th Putnam Mathematical Competition, 1997, Problem A-5

D. Singmaster, The number of representations of one as a sum of unit fractions, Unpublished M.S., 1972

Index entries for sequences related to Egyptian fractions


For n=3 the 10 solutions are {2,3,6} (6 ways), {2,4,4} (3 ways), {3,3,3} (1 way).


Cf. A002966, A006585.

Cf. A000058.

Sequence in context: A213788 A278125 A274763 * A243476 A305107 A294850

Adjacent sequences:  A002964 A002965 A002966 * A002968 A002969 A002970




N. J. A. Sloane


a(7) from Jud McCranie

a(8) from John Dethridge, Jan 11 2004



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Last modified January 20 14:02 EST 2020. Contains 331094 sequences. (Running on oeis4.)