A002966 -id:A002966 - cp's OEIS frontend

A241883 Number of ways 1/n can be expressed as the sum of four distinct unit fractions: 1/n = 1/w + 1/x + 1/y + 1/z satisfying 0 < w < x < y < z.

6, 71, 272, 586, 978, 1591, 1865, 3115, 3772, 4964, 4225, 8433, 4987, 10667, 13659, 10845, 7513, 17360, 9569, 28554, 23309, 17220, 12326, 37554, 19984, 24091, 31056, 42343, 16095, 57001, 15076, 42655, 46885, 38416, 77887, 71959, 16692, 42054, 68894, 95914, 24566, 100023, 24224, 99437, 108756, 41907, 29711, 127069, 52811, 94745, 83433

Offset: 1

Robert G. Wilson v, Apr 30 2014

nonn

			1/1 = 1/2 + 1/3 + 1/7  + 1/42
    = 1/2 + 1/3 + 1/8  + 1/24
    = 1/2 + 1/3 + 1/9  + 1/18
    = 1/2 + 1/3 + 1/10 + 1/15
    = 1/2 + 1/4 + 1/5  + 1/20
    = 1/2 + 1/4 + 1/6  + 1/12
so a(1) = 6.

Jud McCranie, Table of n, a(n) for n = 1..644

Cf. A002966, A004194, A020327, A227610.

a[n_] := Length@ Solve[1/n == 1/w + 1/x + 1/y + 1/z && 0 < w < x < y < z, {w, x, y, z}, Integers]; Array[f, 21]

A092666 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), with 0 < x_1 <= ... <= x_k = n.

Original entry on oeis.org

1, 1, 1, 2, 1, 7, 1, 10, 10, 26, 1, 107, 1, 83, 375, 384, 1, 1418, 1, 4781, 7812, 1529, 1, 33665, 9789, 4276, 27787, 168107, 1, 584667, 1, 586340, 1177696, 52334, 5285597, 14746041, 1, 218959, 13092673, 84854683, 1, 279357910, 1, 491060793, 2001103921

Offset: 1

Max Alekseyev, Mar 02 2004

nonn

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

Toshitaka Suzuki, Table of n, a(n) for n = 1..390
Index entries for sequences related to Egyptian fractions

Cf. A020473, A092667, A092669, A002966, A000058, A259633.

a(n) = A020473(n) - A020473(n-1).

a(n) = 1 if n is prime.

Edited by Max Alekseyev, May 05 2010

A325619 Heinz numbers of integer partitions whose reciprocal factorial sum is 1.

Original entry on oeis.org

2, 9, 375, 15625

Offset: 1

Gus Wiseman, May 13 2019

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      9: {2,2}
    375: {2,3,3,3}
  15625: {3,3,3,3,3,3}

Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.

Reciprocal factorial sum: A002966, A051908, A316855, A325618, A325624.

Select[Range[100000],Total[Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]!]]==1&]

Contains prime(n)^(n!) for all n > 0, including 191581231380566414401 for n = 4.

A325620 Number of integer partitions of n whose reciprocal factorial sum is an integer.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 14, 14, 15, 16, 18, 19, 20, 22, 24, 25, 26, 28, 31, 33, 34, 36, 39, 41, 43, 45, 49, 52, 54, 57, 61, 65, 68, 71, 76, 80, 84, 88, 93, 98, 103, 107, 113

Offset: 1

Gus Wiseman, May 13 2019

The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.

			The initial terms count the following partitions:
  1: (1)
  2: (1,1)
  3: (1,1,1)
  4: (2,2)
  4: (1,1,1,1)
  5: (2,2,1)
  5: (1,1,1,1,1)
  6: (2,2,1,1)
  6: (1,1,1,1,1,1)
  7: (2,2,1,1,1)
  7: (1,1,1,1,1,1,1)
  8: (2,2,2,2)
  8: (2,2,1,1,1,1)
  8: (1,1,1,1,1,1,1,1)
  9: (2,2,2,2,1)
  9: (2,2,1,1,1,1,1)
  9: (1,1,1,1,1,1,1,1,1)

Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.

Reciprocal factorial sum: A002966, A051908, A058360, A316854, A316856, A325618, A325621, A325622.

Table[Length[Select[IntegerPartitions[n],IntegerQ[Total[1/(#!)]]&]],{n,30}]

A325622 Number of integer partitions of n whose reciprocal factorial sum is the reciprocal of an integer.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 3, 3, 2, 2, 3, 3, 3, 5, 4, 4, 3, 3, 4, 6, 3, 4, 5, 5, 5, 6, 3, 7, 6, 5, 6, 6, 6, 5, 6, 8, 5, 7, 5, 4, 8, 7, 7, 7, 7, 9, 9, 9, 10, 12, 6, 12, 8, 10, 7, 14, 10, 8, 11, 11, 12, 11, 10, 10, 12, 14, 11, 10, 9, 10, 12, 10, 15, 14, 11, 10

Offset: 1

Gus Wiseman, May 13 2019

nonn

The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.

			The initial terms count the following partitions:
   1: (1)
   2: (2)
   3: (3)
   4: (4)
   4: (2,2)
   5: (5)
   6: (6)
   6: (3,3)
   7: (7)
   8: (8)
   8: (4,4)
   9: (9)
   9: (5,4)
   9: (3,3,3)
  10: (10)
  10: (5,5)
  11: (11)
  11: (4,4,3)
  11: (3,3,3,2)
  12: (12)
  12: (6,6)
  12: (4,4,4)

Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.

Reciprocal factorial sum: A002966, A316854, A316857, A325618, A325620, A325623.

f:= proc(n) nops(select(proc(t) local i; (1/add(1/i!,i=t))::integer end proc, combinat:-partition(n))) end proc:
map(f, [$1..70]); # Robert Israel, May 09 2024

Table[Length[Select[IntegerPartitions[n],IntegerQ[1/Total[1/(#!)]]&]],{n,30}]

a(n) = my(c=0); forpart(v=n, if(numerator(sum(i=1, #v, 1/v[i]!))==1, c++)); c; \\ Jinyuan Wang, Feb 25 2025

a(61)-a(70) from Robert Israel, May 09 2024

a(71)-a(80) from Jinyuan Wang, Feb 25 2025

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

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

A226646 Number of ways to express m/n as Egyptian fractions in just three terms, that is, m/n = 1/x + 1/y + 1/z satisfying 1 <= x <= y <= z and read by antidiagonals.

Original entry on oeis.org

3, 1, 10, 1, 3, 21, 0, 3, 8, 28, 0, 1, 3, 10, 36, 0, 1, 3, 6, 12, 57, 0, 1, 2, 3, 10, 21, 42, 0, 0, 1, 4, 2, 10, 17, 70, 0, 0, 1, 3, 3, 8, 9, 28, 79, 0, 0, 0, 1, 3, 4, 7, 20, 26, 96, 0, 0, 1, 1, 2, 3, 4, 10, 21, 36, 62, 0, 0, 0, 1, 1, 7, 1, 7, 6, 21, 25, 160, 0, 0, 0, 1, 0, 3, 3, 6, 12, 12, 16, 57, 59

Offset: 1

Allan C. Wechsler and Robert G. Wilson v, Aug 17 2013

See A073101 for the 4/n conjecture due to Erdös and Straus.
The first upper diagonal is 10, 8, 6, 2, 4, 1, 2, 1, 2, 0, 3, 0, 0, 1, 0, 0, 1, 0, 1, 0,... .
The main diagonal is: 3, 3, 3, 3, 3, 3, ... since 1 = 1/2 + 1/3 + 1/6 = 1/2 + 1/4 + 1/4 = 1/3 + 1/3 + 1/3. See A002966(3).
The first lower diagonal is 1, 3, 3, 4, 3, 7, 3, 5, 4, 6, 3, 10, 3, 6, 6, 6, 3, 9, 3, 9, ... .
The antidiagonal sum is 3, 11, 25, 39, 50, 79, 79, 104, 131, 157, 140, 229, 169, 220, 295, 282, ... .

			../n
m/ 1...2...3...4...5...6...7...8...9..10..11...12..13...14...15 =Allocation nbr.
.1 3..10..21..28..36..57..42..70..79..96..62..160..59..136..196 A004194
.2 1...3...8..10..12..21..17..28..26..36..25...57..20...42...81 A226641
.3 1...3...3...6..10..10...9..20..21..21..16...28..11...33...36 A226642
.4 0...1...3...3...2...8...7..10...6..12...9...21...4...17...39 A192787
.5 0...1...2...4...3...4...4...7..12..10...3...17...6...21...21 A226644
.6 0...1...1...3...3...3...1...6...8..10...7...10...1....9...12 A226645
.7 0...0...1...1...2...7...3...2...3...5...2...13...8...10....9 n/a
.8 0...0...0...1...1...3...3...3...1...2...0....8...3....7...19 n/a
.9 0...0...1...1...0...3...2...5...3...2...0....6...2....4...10 n/a
10 0...0...0...1...1...2...0...4...4...3...0....4...1....4....8 n/a
Triangle (by antidiagonals):
  {3},
  {1, 10},
  {1, 3, 21},
  {0, 3, 8, 28},
  {0, 1, 3, 10, 36},
  {0, 1, 3, 6, 12, 57},
  ...

Christian Elsholtz, Sums Of k Unit Fractions
David Eppstein, Algorithms for Egyptian Fractions
David Eppstein, Ten Algorithms for Egyptian Fractions
Ron Knott, Egyptian Fractions
Oakland University, The Erdős Number Project
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A227612, A226640, A226641, A226642, A192787, A226644, A226645.

f[m_, n_] := Length@ Solve[m/n == 1/x + 1/y + 1/z && 1 <= x <= y <= z, {x, y, z}, Integers]; Table[f[n, m - n + 1], {m, 12}, {n, m, 1, -1}] // Flatten

A325623 Heinz numbers of integer partitions whose reciprocal factorial sum is the reciprocal of an integer.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 221, 223, 227, 229

Offset: 1

Gus Wiseman, May 13 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   29: {10}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   47: {15}
   49: {4,4}
   53: {16}

Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.

Reciprocal factorial sum: A002966, A051908, A058360, A316854, A316857, A325619, A325621, A325622.

Select[Range[100],IntegerQ[1/Total[Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]!]]]&]

A348625 Number of Egyptian fractions with squared denominators: number of solutions to the equation 1 = 1/x_1^2 + ... + 1/x_n^2 with 0 < x_1 <= ... <= x_n.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 4, 7, 47, 186, 1809, 27883

Offset: 1

Max Alekseyev, Oct 25 2021

All denominators are bounded by A348626(n), i.e., 0 < x_1 <= ... <= x_n < A348626(n). Furthermore, for a fixed n, x_i <= sqrt(n+1-i)*(A348626(i)-1).

Cf. A002966, A348626.

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

cp's OEIS Frontend

A241883 Number of ways 1/n can be expressed as the sum of four distinct unit fractions: 1/n = 1/w + 1/x + 1/y + 1/z satisfying 0 < w < x < y < z.

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A092666 a(n) = number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k (for any k), with 0 < x_1 <= ... <= x_k = n.

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A325619 Heinz numbers of integer partitions whose reciprocal factorial sum is 1.

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A325620 Number of integer partitions of n whose reciprocal factorial sum is an integer.

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A325622 Number of integer partitions of n whose reciprocal factorial sum is the reciprocal of an integer.

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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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A226646 Number of ways to express m/n as Egyptian fractions in just three terms, that is, m/n = 1/x + 1/y + 1/z satisfying 1 <= x <= y <= z and read by antidiagonals.

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A325623 Heinz numbers of integer partitions whose reciprocal factorial sum is the reciprocal of an integer.

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Mathematica

A348625 Number of Egyptian fractions with squared denominators: number of solutions to the equation 1 = 1/x_1^2 + ... + 1/x_n^2 with 0 < x_1 <= ... <= x_n.