A028229 -id:A028229 - cp's OEIS frontend

A051908 Number of ways to express 1 as the sum of unit fractions such that the sum of the denominators is n.

1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 3, 0, 1, 1, 1, 1, 2, 3, 2, 2, 1, 2, 2, 2, 4, 5, 5, 2, 4, 5, 5, 9, 4, 4, 6, 4, 4, 7, 8, 4, 10, 9, 9, 11, 8, 13, 13, 15, 16, 21, 18, 16, 22, 19, 18, 30, 24, 19, 26, 28, 26, 29, 35, 29, 44, 28, 47, 48

Offset: 1

Jud McCranie, Dec 16 1999

nonn

Also the number of partitions of n whose reciprocal sums to 1; "exact partitions". - Robert G. Wilson v, Sep 30 2009

			1 = 1/2 + 1/2, the sum of denominators is 4, and this is the only expression of 1 as unit fractions with denominator sum 4, so a(4)=1.
The a(22) = 3 partitions whose reciprocal sum is 1 are (12,4,3,3), (10,5,5,2), (8,8,4,2). - _Gus Wiseman_, Jul 16 2018

Derrick Niederman, "Number Freak, From 1 to 200 The Hidden Language of Numbers Revealed", a Perigee Book, Penguin Group, NY, 2009, pp. 82-83. [From Robert G. Wilson v, Sep 30 2009]

David A. Corneth, Table of n, a(n) for n = 1..200 (terms a(1)-a(86) from Jud McCranie, a(87)-a(88) from Robert G. Wilson v, a(89)-a(100) from Seiichi Manyama)
David A. Corneth, Tuples up to n = 170
Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums
Index entries for sequences related to Egyptian fractions

A028229 lists n such that a(n)=0.

Cf. A002966, A058360, A270599, A316854, A316855, A316888-A316904.

(* first do *) << "Combinatorica`"; (* then *) f[n_] := Block[{c = i = 0, k = PartitionsP@n, p = {n}}, While[i < k, If[1 == Plus @@ (1/p), c++ ]; i++; p = NextPartition@p]; c]; Array[f, 88] (* Robert G. Wilson v, Sep 30 2009 *)
Table[Length[Select[IntegerPartitions[n],Sum[1/m,{m,#}]==1&]],{n,30}] (* Gus Wiseman, Jul 16 2018 *)

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def A051908(n)
  ary = [1]
  (2..n).each{|m|
    cnt = 0
    partition(m, 2, m).each{|ary|
      cnt += 1 if ary.inject(0){|s, i| s + 1 / i.to_r} == 1
    }
    ary << cnt
  }
  ary
end
p A051908(100) # Seiichi Manyama, May 31 2016

a(n) > 0 for n > 23.

A051882 Call m strict-sense Egyptian if we can partition m = x_1+x_2+...+x_k into distinct positive integers x_i such that Sum_{i=1..k} 1/x_i = 1; sequence gives all numbers that are not strict-sense Egyptian.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 33, 34, 35, 36, 39, 40, 41, 42, 44, 46, 47, 48, 49, 51, 56, 58, 63, 68, 70, 72, 77

Offset: 1

Jud McCranie, Dec 15 1999

Lehmer shows that 77 is in this sequence. Graham shows that it is the last member of the sequence.

			1=1/2+1/3+1/6, so 2+3+6=11 is strict-sense Egyptian.

D. H. Lehmer, unpublished work, cited in Graham 1963.
See also R. K. Guy, Unsolved Problems Number Theory, Sect. D11.

R. L. Graham, A theorem on partitions, J. Austral. Math. Soc. 3:4 (1963), pp. 435-441. [Alternate link]
Eric Weisstein's World of Mathematics, Egyptian Number
Index entries for sequences related to Egyptian fractions

Cf. A028229.

strictEgyptianQ[m_] := Length[ Select[ IntegerPartitions[m, Ceiling[(Sqrt[8*m + 1] - 1)/2]], Length[#] == Length[ Union[#]] && 1 == Plus @@ (1/#) & , 1]] > 0; Reap[ Do[ If[ !strictEgyptianQ[m], Print[m]; Sow[m]], {m, 1, 77}]][[2, 1]] (* Jean-François Alcover, Jul 30 2012 *)

A052428 Strict Egyptian numbers (complement of A051882).

Original entry on oeis.org

1, 11, 24, 30, 31, 32, 37, 38, 43, 45, 50, 52, 53, 54, 55, 57, 59, 60, 61, 62, 64, 65, 66, 67, 69, 71, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111

Offset: 1

Eric W. Weisstein

Max A. Alekseyev, On partitions into squares of distinct integers whose reciprocals sum to 1, The Mathematics of Various Entertaining Subjects: Volume 3: The Magic of Mathematics, Part III Algebra and Number Theory (2019), 213-221.
R. L. Graham, A theorem on partitions, J. Austral. Math. Soc. 3:4 (1963), pp. 435-441.
Eric Weisstein's World of Mathematics, Egyptian Number.
Index entries for sequences related to Egyptian fractions
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A051882, A028229.

Note that A051909 is a subset (but is strictly different).

strictEgyptianQ[m_]:=!Length[Select[IntegerPartitions[m, Ceiling[(Sqrt[8 m + 1] - 1) / 2]], Length[#]==Length[Union[#]]&&1==Plus@@(1/#)&, 1]]>0; Reap[Do[If[!strictEgyptianQ[m], Print[m]; Sow[m]], {m, 1,100}]][[2, 1]] (* Vincenzo Librandi, Jul 16 2017 *)

a(n) = n + 47 for n > 30. - Charles R Greathouse IV, Jun 19 2024

A125726 Call n Egyptian if we can partition n = x_1+x_2+...+x_k into positive integers x_i such that Sum_{i=1..k} 1/x_i = 1; sequence gives Egyptian numbers.

Original entry on oeis.org

1, 4, 9, 10, 11, 16, 17, 18, 20, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86

Offset: 1

Jan RUCKA (jan_rucka(AT)hotmail.com), Feb 06 2007

nonn

			1=1/3+1/3+1/3, so 3+3+3=9 is Egyptian.

J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 147.
See also R. K. Guy, Unsolved Problems Number Theory, Sect. D11.

R. L. Graham, A theorem on partitions, J. Austral. Math. Soc., 4 (1963), 435-441.
Les Mathematiques.net, Nombres remarquables (French blog).
Eric Weisstein's World of Mathematics, Egyptian Number.
Index entries for sequences related to Egyptian fractions

Complement of A028229.

cp's OEIS Frontend

A051908 Number of ways to express 1 as the sum of unit fractions such that the sum of the denominators is n.

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A051882 Call m strict-sense Egyptian if we can partition m = x_1+x_2+...+x_k into distinct positive integers x_i such that Sum_{i=1..k} 1/x_i = 1; sequence gives all numbers that are not strict-sense Egyptian.

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A052428 Strict Egyptian numbers (complement of A051882).

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Programs

Mathematica

Formula

A125726 Call n Egyptian if we can partition n = x_1+x_2+...+x_k into positive integers x_i such that Sum_{i=1..k} 1/x_i = 1; sequence gives Egyptian numbers.

Views

Author

Keywords

Examples

References

Links

Crossrefs