A006585 Egyptian fractions: number of solutions to 1 = 1/x_1 + ... + 1/x_n in positive integers x_1 < ... < x_n. (Formerly M4281)

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

Offset

1,4

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

References

Marc LeBrun, personal communication.

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.

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

Formula

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

Example

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.

Crossrefs

Cf. A000058, A002966, A002967, A280518.

Sequence in context: A259212 A279234 A132878 * A166472 A182917 A328814

Adjacent sequences: A006582 A006583 A006584 * A006586 A006587 A006588

Keyword

nonn,nice,hard,more

Author

N. J. A. Sloane

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

Status

approved