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
A156869
Triangle read by rows: T(n,k) = number of nondecreasing sequences of n positive integers with reciprocals adding up to k (1 <= k <= n).
Original entry on oeis.org
1, 1, 1, 3, 1, 1, 14, 4, 1, 1, 147, 17, 4, 1, 1, 3462, 164, 18, 4, 1, 1, 294314, 3627, 167, 18, 4, 1, 1, 159330691, 297976, 3644, 168, 18, 4, 1, 1
Offset: 1
Triangle begins:
n=1: 1
n=2: 1, 1
n=3: 3, 1, 1
n=4: 14, 4, 1, 1
n=5: 147, 17, 4, 1, 1
n=6: 3462, 164, 18, 4, 1, 1
n=7: 294314, 3627, 167, 18, 4, 1, 1
For n = 4 and k = 2, the T(4, 2) = 4 sequences are (1, 2, 3, 6), (1, 2, 4, 4), (1, 3, 3, 3) and (2, 2, 2, 2) because 1/1 + 1/2 + 1/3 + 1/6 = 1/1 + 1/2 + 1/4 + 1/4 = 1/1 + 1/3 + 1/3 + 1/3 = 1/2 + 1/2 + 1/2 + 1/2 = 2.
-
{ A156869(n,k,m=1) = n==1 & return(numerator(k)==1 & denominator(k)>=m); sum( i=max(m,1\k+1),n\k, A156869(n-1, k-1/i, i)); } \\ M. F. Hasler, Feb 20 2009
A280518
Number of increasing sequences of n positive integers with reciprocals adding up to an integer.
Original entry on oeis.org
1, 0, 1, 7, 78, 2392, 248085, 151428144
Offset: 1
a(4)=7 enumerates sequences: (1,2,3,6), (2,4,5,20), (2,4,6,12), (2,3,7,42), (2,3,8,24), (2,3,9,18), and (2,3,10,15).
A280519
Triangle read by rows: T(n,k) = number of sequences of n positive integers with reciprocals adding up to k (k=1,2,...,n).
Original entry on oeis.org
1, 1, 1, 10, 3, 1, 215, 41, 6, 1, 12231, 1115, 105, 10, 1, 2025462, 74862, 3466, 215, 15, 1, 1351857641, 14294210, 267281, 8372, 385, 21, 1, 6255560531733, 10837663111, 57646358, 727049, 17318, 630, 28, 1
Offset: 1
Triangle starts with:
n=1: 1
n=2: 1, 1
n=3: 10, 3, 1
n=4: 215, 41, 6, 1
n=5: 12231, 1115, 105, 10, 1
n=6: 2025462, 74862, 3466, 215, 15, 1
...
Showing 1-4 of 4 results.
Comments