A002967 -id:A002967 - 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D. Singmaster, The number of representations of one as a sum of unit fractions, unpublished manuscript, 1972.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David Singmaster, The number of representations of one as a sum of unit fractions, Unpublished M.S., 1972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

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

N. J. A. Sloane

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

If on the other hand, x_k need not be unique, see A002966. - Robert G. Wilson v, Jul 17 2013

			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

Cf. A000058, A002966, A002967, A280518.

a(n) = A280520(n,1).

a(1)-a(7) are confirmed by Jud McCranie, Dec 11 1999

a(8) from John Dethridge (jcd(AT)ms.unimelb.edu.au), Jan 08 2004

A038034 Number of compositions (ordered partitions) of 1 into {1/1, 1/2, 1/3, ..., 1/n}.

Original entry on oeis.org

1, 2, 3, 7, 8, 52, 53, 288, 1209, 5247, 5248, 71395, 71396, 375779, 6957533, 52310862, 52310863, 1152622553, 1152622554, 45575902465, 1296407854551, 1580527987951, 1580527987952, 73245316681199, 584407520822198, 639887219617512, 11355804443049274, 516959218512416104, 516959218512416105, 29213061562205847736, 29213061562205847737, 886912328033731357358, 31286298736622399674197, 31349361777225437765677

Offset: 1

Christian G. Bower, Jun 15 1998

a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), with max{x_i}<=n.

			a(4) = 7 since there are seven compositions into parts {1/1, 1/2, 1/3, 1/4}:
1 = 1/1, 1 = 1/2 + 1/2, 1 = 1/3 + 1/3 + 1/3, 1 = 1/2 + 1/4 + 1/4, 1 = 1/4 + 1/2 + 1/4, 1 = 1/4 + 1/4 + 1/2, and 1 = 1/4 + 1/4 + 1/4 + 1/4.

Index entries for sequences related to Egyptian fractions

Cf. A002967, A020473, A092667, A092670.

a(n) = Sum_{i=1..n} A092667(i).

a(p) = a(p-1) + 1 for p prime. - Chai Wah Wu, Dec 27 2024

More terms from Max Alekseyev, Mar 02 2004

A092667 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), with max{x_i}=n.

Original entry on oeis.org

1, 1, 1, 4, 1, 44, 1, 235, 921, 4038, 1, 66147, 1, 304383, 6581754, 45353329, 1, 1100311690, 1, 44423279911, 1250831952086, 284120133400, 1, 71664788693247, 511162204140999, 55479698795314, 10715917223431762, 505603414069366830, 1, 28696102343693431631, 1, 857699266471525509621, 30399386408588668316839, 63063040603038091480

Offset: 1

Max Alekseyev, Mar 02 2004

nonn

			a(4) = 4 since there are four fractions 1=1/2+1/4+1/4, 1=1/4+1/2+1/4, 1=1/4+1/4+1/2 and 1=1/4+1/4+1/4+1/4.

Index entries for sequences related to Egyptian fractions

Cf. A092666, A038034, A092669, A002967, A000058.

a(n) = A038034(n) - A038034(n-1).

a(n) = 1 if n is prime.

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).

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).

A144063 Egyptian fractions: number of solutions of 1 = 1/x_1 + ... + 1/x_n in positive integers x_1 < ... < x_n <= 256.

Original entry on oeis.org

1, 0, 1, 6, 62, 642, 5623, 47126, 368680, 2715613, 18876751, 124137535, 774232619, 4595291801, 26030660449, 141031079451, 731862267491, 3641135367129, 17379359388167, 79633646141291, 350541406992141, 1483638948734104, 6043258919626951, 23713645892145709

Offset: 1

Zhao Hui Du, Sep 09 2008

Egyptian fraction for a rational number is to represent the number in sum of some distinct unit fraction, such as 1 = 1/2 + 1/3 + 1/6, here we represent 1 by a three terms Egyptian fraction and the largest denominator is 6. And the representation is non-unique.

There are a total of 3007198863516917545589795267613 Egyptian fractions for 1 whose largest denominator is no more than 256.

			a(1)=1 since 1 = 1/1.
a(2)=0 since the sum of any two distinct unit fractions are not 1.
a(3)=1 since the only three terms Egyptian fraction for 1 is 1/2 + 1/3 + 1/6.

Jinyuan Wang, Table of n, a(n) for n = 1..114
Eric Weisstein's World of Mathematics, Egyptian Fraction
link for the result [Broken link]

Cf. A002967, A006585.

a(n) = 0 for n > 114.

A343214 Number of compositions (ordered partitions) of 1/n into n reciprocals of positive integers.

Original entry on oeis.org

1, 3, 106, 15259, 13879541, 95073863568

Offset: 1

Ilya Gutkovskiy, Apr 08 2021

			a(2) = 3 because we have 1/2 = 1/4 + 1/4 = 1/3 + 1/6 = 1/6 + 1/3.

Cf. A002967, A048691, A342267, A343074.

a(5)-a(6) from Max Alekseyev, Jan 24 2024

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

Max Alekseyev, Jul 12 2024

Also, number of integers m such that m is the sum of n distinct divisors of m including 1.

x_n <= A000058(n-1)-1.

Cf. A002967, A280518, A374582.

A379452 Number of compositions (ordered partitions) of 1 into n distinct reciprocals of positive integers.

Original entry on oeis.org

1, 0, 6, 144, 8640, 1670400, 1238655600, 6095673521280

Offset: 1

Ilya Gutkovskiy, Dec 23 2024

			a(3) = 6 because we have 1 = 1/2 + 1/3 + 1/6
                           = 1/2 + 1/6 + 1/3
                           = 1/3 + 1/2 + 1/6
                           = 1/3 + 1/6 + 1/2
                           = 1/6 + 1/2 + 1/3
                           = 1/6 + 1/3 + 1/2.

Cf. A002966, A002967, A006585.

a(n) = n! * A006585(n).

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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A006585 Egyptian fractions: number of solutions to 1 = 1/x_1 + ... + 1/x_n in positive integers x_1 < ... < x_n.

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A038034 Number of compositions (ordered partitions) of 1 into {1/1, 1/2, 1/3, ..., 1/n}.

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A092667 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), with max{x_i}=n.

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

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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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A144063 Egyptian fractions: number of solutions of 1 = 1/x_1 + ... + 1/x_n in positive integers x_1 < ... < x_n <= 256.

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A343214 Number of compositions (ordered partitions) of 1/n into n reciprocals of positive integers.

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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.

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A379452 Number of compositions (ordered partitions) of 1 into n distinct reciprocals of positive integers.

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