A280519 -id:A280519 - cp's OEIS frontend

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).

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

Jens Voß, Feb 17 2009

Conjecture: T(2n + m, n + m) = T(2n, n) ( = A156870(n) ) if and only if m >= 0.
Yes, the diagonals are constant for n <= 2k. Any such sequence must have at least one 1; remove that 1, and you get a sequence for n-1,k-1. - Franklin T. Adams-Watters, Feb 20 2009
The next term will be a(37) = A002966(9). - M. F. Hasler, Feb 20 2009

			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.

Cf. A002966 (column k=1), A156871 (row sums), A280519, A280520.

T(2n, n) = A156870(n).

{ 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

a(21)-a(36) from M. F. Hasler, Feb 20 2009

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

Max Alekseyev, Jan 04 2017

T(n,k) = 0 for all k > A055980(n).
For n=3,...,11, we have T(n,2) = T(n-1,1). However, T(12,2) > T(11,1).
Conjecture: for n in A115515 (i.e., A055980(n+1)=A055980(n)+1), the sequences being enumerated by T(n,A055980(n)) must start with 1. E.g., there is no 10-tuple (x_1,x_2,...,x_10) with 1 < x_1 < ... < x_10 and 1/x_1 + ... + 1/x_10 = 2 (=A055980(10)).

			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
...

Cf. A280518 (row sums), A006585 (column k=1), A156869 (nondecreasing sequences), A280519 (ordered sequences).

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

Max Alekseyev, Jan 04 2017

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.

Row sums of A280519.

Cf. A002967 (adding up to 1), A156871 (nondecreasing sequences), A280518 (increasing sequences).

cp's OEIS Frontend

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).

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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)).

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A280517 Number of sequences of n positive integers with reciprocals adding up to an integer.

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