A156869 -id:A156869 - cp's OEIS frontend

A002966 Egyptian fractions: number of solutions of 1 = 1/x_1 + ... + 1/x_n where 0 < x_1 <= ... <= x_n.

1, 1, 3, 14, 147, 3462, 294314, 159330691

Offset: 1

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
From R. J. Mathar, May 06 2010: (Start)
This is the leading edge of the triangle A156869. This is also the row n=1 of an array T(n,m) which gives the number of ways to write 1/n as a sum over m (not necessarily distinct) unit fractions:
   1, 1,  3,  14,   147,   3462, 294314, ...
   1, 2, 10, 108,  2892, 270332, ...
   1, 2, 21, 339, 17253, ...
   1, 3, 28, 694, 51323, ...
   ...
T(.,2) = A018892. T(.,3) = A004194. T(.,4) = A020327, T(.,5) = A020328. T(2,6) is computed by D. S. McNeil, who conjectures that the 2nd row is A003167. (End)
If on the other hand, all x_k must be unique, see A006585. - Robert G. Wilson v, Jul 17 2013

			For n=3 the 3 solutions are {2,3,6}, {2,4,4}, {3,3,3}.
For n=4 the solutions are: {2,3,7,42}, {2,3,8,24}, {2,3,9,18}, {2,3,10,15}, {2,3,12,12}, {2,4,5,20}, {2,4,6,12}, {2,4,8,8}, {2,5,5,10}, {2,6,6,6}, {3,3,4,12}, {3,3,6,6}, {3,4,4,6}, {4,4,4,4}. [Neven Juric, May 14 2008]

R. K. Guy, Unsolved Problems in Number Theory, D11.
D. Singmaster, The number of representations of one as a sum of unit fractions, unpublished manuscript, 1972.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Matthew Brendan Crawford, On the Number of Representations of One as the Sum of Unit Fractions, Master's Thesis, Virginia Polytechnic Institute and State University (2019).
Yuya Dan, Representation of one as the sum of unit fractions, International Mathematical Forum 6:1 (2011), pp. 25-30.
Jacques Le Normand, C++ code for a(8) [Broken link]
Jacques Le Normand, C++ code for a(8) [Cached copy]
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.
David Singmaster, The number of representations of one as a sum of unit fractions, Unpublished M.S., 1972
R. G. Wilson, v, Fax to N. J. A. Sloane, Sep 9, 1994, with copy of Scientific American column by Ian Stewart
Index entries for sequences related to Egyptian fractions

Cf. A002967, A006585, A000058, A348625.

a(n,rem=1,mn=1)=if(n==1,return(numerator(rem)==1)); sum(k=max(1\rem+1,mn), n\rem, a(n-1,rem-1/k,k)) \\ Charles R Greathouse IV, Jan 04 2015

a(n) <= binomial(A007018(n), n-1). - Charles R Greathouse IV, Jul 29 2024

a(7) from Jud McCranie, Nov 15 1999. Confirmed by Marc Paulhus.

a(8) from John Dethridge (jcd(AT)ms.unimelb.edu.au) and Jacques Le Normand (jacqueslen(AT)sympatico.ca), Jan 06 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

Jens Voß, Feb 17 2009

			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.

Cf. A002966, A156869, A280517, A280518.

a(n) = A156869(n, 1) + ... + A156869(n, n).

a(7), a(8) from Max Alekseyev, Jul 27 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).

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

Max Alekseyev, Jan 04 2017

			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
...

Cf. A280517 (row sums), A002967 (column k=1), A156869 (nondecreasing sequences), A280520 (increasing sequences).

A156870 Number of nondecreasing sequences of 2n positive integers with reciprocals adding up to n (1 <= n).

Original entry on oeis.org

1, 4, 18, 168, 3648, 298165, 159628076

Offset: 1

Jens Voß, Feb 17 2009

Conjecture: A156869(2n + m, n + m) = a(n) if and only if m >= 0.

			For n = 2, the a(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.

a(n) = A156869(2n, n).

a(6) from M. F. Hasler, Feb 21 2009

a(7) from Max Alekseyev, Jan 04 2017

cp's OEIS Frontend

A002966 Egyptian fractions: number of solutions of 1 = 1/x_1 + ... + 1/x_n where 0 < x_1 <= ... <= x_n.

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A156871 Number of nondecreasing sequences of n positive integers with reciprocals adding up to an integer.

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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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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).

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A156870 Number of nondecreasing sequences of 2n positive integers with reciprocals adding up to n (1 <= n).

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