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.
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{ 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).
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.
A374582
a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_n such that 1 <= x_1 <= ... <= x_n and x_k | x_n for all k = 1..n.
Original entry on oeis.org
1, 1, 3, 12, 97, 1568, 76309, 16993752
Offset: 1
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