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

1,22

Comments

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

References

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]

Links

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

Formula

a(n) > 0 for n > 23.

Example

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

Mathematica

(* 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 *)

Prog

(Ruby)
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

Crossrefs

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

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

Sequence in context: A338939 A292150 A181875 * A056614 A126309 A338940

Adjacent sequences: A051905 A051906 A051907 * A051909 A051910 A051911

Keyword

nonn

Author

Jud McCranie, Dec 16 1999

Status

approved