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
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
KEYWORD
nonn
AUTHOR
Jud McCranie, Dec 16 1999
STATUS
approved