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

Offset

1,3

Comments

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

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D. Singmaster, "The number of representations of one as a sum of unit fractions," unpublished manuscript, 1972.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

Zachary Harris and Joel Louwsma, On Arithmetical Structures on Complete Graphs, arXiv:1909.02022 [math.NT], 2019. See also Involve (2020) Vol. 13, No. 2, 345-355.

Claire Levaillant, On arithmetical structures on K9, arXiv:2311.03668 [math.NT], 2023.

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D. Singmaster, The number of representations of one as a sum of unit fractions, Unpublished M.S., 1972.

Carlos E. Valencia and R. R. Villagrán, Algorithmic aspects of arithmetical structures, arXiv:2101.05238 [math.NT], 2021.

Index entries for sequences related to Egyptian fractions

Example

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

Crossrefs

Cf. A002966, A006585.

Cf. A000058.

Sequence in context: A213788 A278125 A274763 * A243476 A305107 A294850

Adjacent sequences: A002964 A002965 A002966 * A002968 A002969 A002970

Keyword

nonn,nice,hard,more

Author

N. J. A. Sloane

Extensions

a(7) from Jud McCranie

a(8) from John Dethridge, Jan 11 2004

Status

approved