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

%I A051908 #48 Oct 24 2018 08:21:06
%S A051908 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,
%T A051908 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,
%U A051908 22,19,18,30,24,19,26,28,26,29,35,29,44,28,47,48
%N A051908 Number of ways to express 1 as the sum of unit fractions such that the sum of the denominators is n.
%C A051908 Also the number of partitions of n whose reciprocal sums to 1; "exact partitions". - _Robert G. Wilson v_, Sep 30 2009
%D A051908 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]
%H A051908 David A. Corneth, <a href="/A051908/b051908.txt">Table of n, a(n) for n = 1..200</a> (terms a(1)-a(86) from Jud McCranie, a(87)-a(88) from Robert G. Wilson v, a(89)-a(100) from Seiichi Manyama)
%H A051908 David A. Corneth, <a href="/A051908/a051908.gp.txt">Tuples up to n = 170</a>
%H A051908 Gus Wiseman, <a href="/A051908/a051908.txt">Sequences counting and ranking integer partitions by their reciprocal sums</a>
%H A051908 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%F A051908 a(n) > 0 for n > 23.
%e A051908 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.
%e A051908 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
%t A051908 (* 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 *)
%t A051908 Table[Length[Select[IntegerPartitions[n],Sum[1/m,{m,#}]==1&]],{n,30}] (* _Gus Wiseman_, Jul 16 2018 *)
%o A051908 (Ruby)
%o A051908 def partition(n, min, max)
%o A051908   return [[]] if n == 0
%o A051908   [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
%o A051908 end
%o A051908 def A051908(n)
%o A051908   ary = [1]
%o A051908   (2..n).each{|m|
%o A051908     cnt = 0
%o A051908     partition(m, 2, m).each{|ary|
%o A051908       cnt += 1 if ary.inject(0){|s, i| s + 1 / i.to_r} == 1
%o A051908     }
%o A051908     ary << cnt
%o A051908   }
%o A051908   ary
%o A051908 end
%o A051908 p A051908(100) # _Seiichi Manyama_, May 31 2016
%Y A051908 A028229 lists n such that a(n)=0.
%Y A051908 Cf. A002966, A058360, A270599, A316854, A316855, A316888-A316904.
%K A051908 nonn
%O A051908 1,22
%A A051908 _Jud McCranie_, Dec 16 1999