A002967 Egyptian fractions: number of solutions of 1 = 1/x_1 + ... + 1/x_n in positive integers.

1, 1, 10, 215, 12231, 2025462, 1351857641, 6255560531733

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard
• more

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

Examples

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

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• D. Singmaster, "The number of representations of one as a sum of unit fractions," unpublished manuscript, 1972.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

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• D. Singmaster, The number of representations of one as a sum of unit fractions, Unpublished M.S., 1972.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

Crossrefs

Cf. A002966, A006585.

Cf. A000058.

Extensions

a(7) from Jud McCranie

a(8) from John Dethridge, Jan 11 2004