A006585 Egyptian fractions: number of solutions to 1 = 1/x_1 + ... + 1/x_n in positive integers x_1 < ... < x_n.

1, 0, 1, 6, 72, 2320, 245765, 151182379

Offset: 1

Author

N. J. A. Sloane

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

If on the other hand, x_k need not be unique, see A002966. - Robert G. Wilson v, Jul 17 2013

Examples

			The 6 solutions for n=4 are 2,3,7,42; 2,3,8,24; 2,3,9,18; 2,3,10,15; 2,4,5,20; 2,4,6,12.
		

References

• Marc LeBrun, personal communication.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• M. Le Brun, Email to N. J. A. Sloane, Jul 1991
• S. V. Konyagin, Double exponential lower bound for the number of representations of unity by Egyptian fractions. Mathematical Notes, 95:1-2 (2014), 277-281.
• T. D. Browning and C. Elsholtz, The number of representations of rationals as a sum of unit fractions, Illinois J. Math. 55:2 (2011), 685-696.
• 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.
• Index entries for sequences related to Egyptian fractions

Crossrefs

Cf. A000058, A002966, A002967, A280518.

Formula

a(n) = A280520(n,1).

Extensions

a(1)-a(7) are confirmed by Jud McCranie, Dec 11 1999

a(8) from John Dethridge (jcd(AT)ms.unimelb.edu.au), Jan 08 2004