A006585 -id:A006585 - 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D. Singmaster, The number of representations of one as a sum of unit fractions, unpublished manuscript, 1972.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David Singmaster, The number of representations of one as a sum of unit fractions, Unpublished M.S., 1972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

A002967 Egyptian fractions: number of solutions of 1 = 1/x_1 + ... + 1/x_n in positive integers.

Original entry on oeis.org

1, 1, 10, 215, 12231, 2025462, 1351857641, 6255560531733

Offset: 1

N. J. A. Sloane

Solutions differing only in the order of the x_i are counted as distinct.

All denominators in the expansion 1 = 1/x_1 + ... + 1/x_n are bounded by A000058(n-1) = A129871(n). - Max Alekseyev, Dec 30 2003

			For n=3 the 10 solutions are {2,3,6} (6 ways), {2,4,4} (3 ways), {3,3,3} (1 way).

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).

Zachary Harris and Joel Louwsma, On Arithmetical Structures on Complete Graphs, arXiv:1909.02022 [math.NT], 2019. See also Involve (2020) Vol. 13, No. 2, 345-355.
Claire Levaillant, On arithmetical structures on K9, arXiv:2311.03668 [math.NT], 2023.
Putnam Competition, 58th Putnam Mathematical Competition, 1997, Problem A-5
D. Singmaster, The number of representations of one as a sum of unit fractions, Unpublished M.S., 1972.
Carlos E. Valencia and R. R. Villagrán, Algorithmic aspects of arithmetical structures, arXiv:2101.05238 [math.NT], 2021.
Index entries for sequences related to Egyptian fractions

Cf. A002966, A006585.

Cf. A000058.

a(7) from Jud McCranie

a(8) from John Dethridge, Jan 11 2004

A092669 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), 0

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 5, 0, 11, 0, 0, 0, 19, 0, 0, 0, 73, 0, 86, 0, 0, 163, 0, 203, 286, 0, 0, 0, 803, 0, 1399, 0, 0, 2723, 0, 0, 4870, 0, 0, 0, 8789, 0, 13937, 14987, 42081, 0, 0, 0, 85577, 0, 0, 159982, 0, 117889, 437874, 0, 0, 0, 818640, 0

Offset: 1

Max Alekseyev, Mar 02 2004

nonn

For a given n, the Mathematica program uses backtracking to count the solutions. The solutions can be printed by uncommenting the print statement. It is very time-consuming for large n. A092671 gives the n that yield a(n) > 0. - T. D. Noe, Mar 26 2004

			a(6) = 1 since there is the only fraction 1 = 1/2+1/3+1/6.

Toshitaka Suzuki, Table of n, a(n) for n = 1..610
Harry Ruderman and Paul Erdős, Problem E2427: Bounds of Egyptian fraction partitions of unity, Amer. Math. Monthly, Vol. 81, No. 7 (1974), 780-782.
Index entries for sequences related to Egyptian fractions

Cf. A092666, A092667, A092670, A092671, A092672, A006585.

n=20; try2[lev_, s_] := Module[{nmim, nmax, si, i}, AppendTo[soln, 0]; If[lev==1, nmin=2, nmin=1+soln[[ -2]]]; nmax=n-1; Do[If[iT. D. Noe, Mar 26 2004 *)

a(n) = A092670(n) - A092670(n-1).

More terms from T. D. Noe, Mar 26 2004

More terms from T. Suzuki (suzuki(AT)scio.co.jp), Nov 24 2006

A092670 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), 0

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 6, 6, 6, 11, 11, 22, 22, 22, 22, 41, 41, 41, 41, 114, 114, 200, 200, 200, 363, 363, 566, 852, 852, 852, 852, 1655, 1655, 3054, 3054, 3054, 5777, 5777, 5777, 10647, 10647, 10647, 10647, 19436, 19436, 33373, 48360, 90441

Offset: 1

Max Alekseyev, Mar 02 2004

nonn

			a(6)=2 since there are two fractions 1=1/1 and 1=1/2+1/3+1/6.

Toshitaka Suzuki, Table of n, a(n) for n = 1..610
Index entries for sequences related to Egyptian fractions

Cf. A020473, A038034, A092669, A006585.

a(n)=Sum(A092669(i), i=1..n)

More terms from T. Suzuki (suzuki(AT)scio.co.jp), Nov 24 2006

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

Max Alekseyev, Jan 04 2017

			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).

Row sums of A280520.

Cf. A006585 (adding up to 1), A156871 (nondecreasing sequences), A280517 (ordered sequences).

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).

A343074 Number of partitions of 1/n into n distinct reciprocals of positive integers.

Original entry on oeis.org

1, 1, 15, 586, 112535, 131223239

Offset: 1

Ilya Gutkovskiy, Apr 04 2021

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

Index entries for sequences related to Egyptian fractions

Cf. A006585, A063647, A227610, A241883, A342267.

a(5) from Bert Dobbelaere, Apr 05 2021

a(6) from Jud McCranie, Sep 03 2021

A369469 a(n) = number of integer solutions to 1 <= x1 < x2 < ... < xn to 1/x1 + ... + 1/xn = (1 - 1/x1) * ... * (1 - 1/xn).

Original entry on oeis.org

1, 1, 1, 24, 293, 9219, 787444

Offset: 1

Max Alekseyev, Jan 23 2024

For any n, A369470(n) >= a(n) >= 1 (see A369607).

Max Muller et al., The diophantine equation \sum_{n=1}^{N} \frac{1}{x_{n}} = \prod_{k=1}^{N} \left(1-\frac{1}{x_{k}} \right), MathOverflow, 2024.

Cf. A006585, A007850, A054377, A085098, A118086, A158649, A164014, A343074, A369470, A369607.

A075461 List of solutions to the Znám problem sorted first by length, then lexicographically.

Original entry on oeis.org

2, 3, 7, 47, 395, 2, 3, 11, 23, 31, 2, 3, 7, 43, 1823, 193667, 2, 3, 7, 47, 403, 19403, 2, 3, 7, 47, 415, 8111, 2, 3, 7, 47, 583, 1223, 2, 3, 7, 55, 179, 24323, 2, 3, 7, 43, 1807, 3263447, 2130014000915, 2, 3, 7, 43, 1807, 3263591, 71480133827, 2, 3, 7, 43

Offset: 1

Eric W. Weisstein, Sep 16 2002

			Starts with A075441(5)=2 5-term solutions 2,3,7,47,395; 2,3,11,23,31, followed by A075441(6)=5 6-term solutions, etc.

Max Alekseyev, Table of n, a(n) for n = 1..934 (flattened list of the n-term solutions for n = 5..8)
J. Janák and L. Skula, On the integers xi for which xi | x1 . . . xi-1 xi . . . xn + 1 holds, Mathematica Slovaca, Vol. 28 (1978), No. 3, 305--310.
J. Sondow and K. MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the Erdos-Moser equation, Amer. Math. Monthly, 124 (2017) 232-240; arXiv:math/1812.06566 [math.NT], 2018.
Eric Weisstein's World of Mathematics, Znám's Problem
Wikipedia, Znám's problem.

Cf. A006585, A007850, A054377, A075441, A085098, A118086, A158649, A164014, A343074, A369469.

Edited by Max Alekseyev, Jan 25 2024

A181700 Smallest positive integer that can be represented as the sum of n of its distinct divisors in the maximum number of ways, or a(n)=0 if no such number exists.

Original entry on oeis.org

1, 0, 6, 2520, 48348686786400, 10543141534556403817127800577537146514577188497111149855093902265479066128013109211427715400552367011213513440000

Offset: 1

William Rex Marshall, Nov 06 2010

A006585(n) gives the number of representations of a(n) as the sum of n of its distinct divisors.

For n >= 4, a(n) appears to coincide with A357765(n). - Max Alekseyev, Oct 09 2022

			For n=4, a(4)=2520 has the six solutions: 60+360+840+1260, 105+315+840+1260, 126+504+630+1260, 140+280+840+1260, 168+252+840+1260, 210+420+630+1260.

Max Alekseyev, Table of n, a(n) for n = 1..7

Cf. A002966, A006585, A357765.

For n >= 3, a(n) = LCM of all denominators of Egyptian fractions enumerated by A006585(n). - Max Alekseyev, Oct 12 2022

Edited by Max Alekseyev, Oct 09 2022

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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A002967 Egyptian fractions: number of solutions of 1 = 1/x_1 + ... + 1/x_n in positive integers.

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A092669 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), 0

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A092670 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), 0

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A280518 Number of increasing 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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A343074 Number of partitions of 1/n into n distinct reciprocals of positive integers.

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A369469 a(n) = number of integer solutions to 1 <= x1 < x2 < ... < xn to 1/x1 + ... + 1/xn = (1 - 1/x1) * ... * (1 - 1/xn).

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A075461 List of solutions to the Znám problem sorted first by length, then lexicographically.

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