A006585
Egyptian fractions: number of solutions to 1 = 1/x_1 + ... + 1/x_n in positive integers x_1 < ... < x_n.
Original entry on oeis.org
1, 0, 1, 6, 72, 2320, 245765, 151182379
Offset: 1
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.
- Marc LeBrun, personal communication.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- 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
a(8) from John Dethridge (jcd(AT)ms.unimelb.edu.au), Jan 08 2004
A156871
Number of nondecreasing sequences of n positive integers with reciprocals adding up to an integer.
Original entry on oeis.org
1, 2, 5, 20, 170, 3650, 298132, 159632503
Offset: 1
For n = 3, the A156871(3) = 5 sequences are (1, 1, 1), (1, 2, 2), (2, 3, 6), (2, 4, 4) and (3, 3, 3) because 1/1 + 1/1 + 1/1 = 3, 1/1 + 1/2 + 1/2 = 2 and 1/2 + 1/3 + 1/6 = 1/2 + 1/4 + 1/4 = 1/3 + 1/3 + 1/3 = 1.
A280520
Triangle read by rows: T(n,k) = number of increasing sequences of n positive integers with reciprocals adding up to k (k=1,2,...,A055980(n)).
Original entry on oeis.org
1, 0, 1, 6, 1, 72, 6, 2320, 72, 245765, 2320, 151182379, 245765
Offset: 1
Triangle starts with:
n=1: 1
n=2: 0
n=3: 1
n=4: 6, 1
n=5: 72, 6
n=6: 2320, 72
n=7: 245765, 2320
n=8: 151182379, 245765
...
A280517
Number of sequences of n positive integers with reciprocals adding up to an integer.
Original entry on oeis.org
1, 2, 14, 263, 13462, 2104021, 1366427911, 6266456586228
Offset: 1
- Kassie Archer, Abigail Bishop, Alexander Diaz-Lopez, Luis David Garcia Puente, Darren Glass, Joel Louwsma, Arithmetical structures on bidents, arXiv:1903.01393 [math.CO], 2019.
- Dilli Ram Chhetri, Namita Behera, Raj Bhawan Yadav, Arithmetical Structures On Fan Graphs, arXiv:2503.01913 [math.CO], 2025. See p. 17.
A374583
a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_n such that 0 < x_1 < ... < x_n and x_k | x_n for all k = 1..n.
Original entry on oeis.org
1, 0, 1, 5, 44, 975, 59234, 15474226
Offset: 1
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