

A051908


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


43



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
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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. 8283. [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, A316888A316904.
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



