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Number of integer partitions of n whose reciprocal sum is the reciprocal of an integer.
31

%I #20 Sep 03 2018 16:30:55

%S 1,1,1,2,1,1,1,2,3,2,2,2,1,1,1,4,2,4,1,5,1,5,1,3,4,2,5,6,5,5,4,5,5,4,

%T 8,10,9,7,5,9,10,6,12,10,8,7,6,9,13,15,8,19,13,19,19,19,18,22,26,28,

%U 28,29,22,33,29,28,38,34,26,40,32,43,39,51,38,62,46

%N Number of integer partitions of n whose reciprocal sum is the reciprocal of an integer.

%C The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

%H Giovanni Resta, <a href="/A316854/b316854.txt">Table of n, a(n) for n = 1..200</a> (first 100 terms from Robert G. Wilson v)

%e The a(36) = 10 partitions:

%e (36),

%e (30,6), (24,12), (18,18),

%e (12,12,12),

%e (12,12,6,6),

%e (15,10,4,4,3), (12,12,6,3,3), (12,8,8,6,2),

%e (6,6,6,6,6,6).

%t Table[Length[Select[IntegerPartitions[n],IntegerQ[1/Sum[1/m,{m,#}]]&]],{n,30}]

%t ric[n_, p_, s_] := If[n == 0, If[IntegerQ[1/s], c++], Do[If[s + 1/i <= 1, ric[n - i, Append[p, i], s + 1/i]], {i, Min[p[[-1]], n], 1, -1}]]; a[n_] := (c = 0; Do[ric[n - j, {j}, 1/j], {j, n}]; c); Array[a, 80] (* after Giovanni Resta in A316898, _Robert G. Wilson v_, Jul 23 2018 *)

%o (PARI) a(n)={my(s=0); forpart(p=n, if(frac(1/sum(i=1, #p, 1/p[i]))==0, s++)); s} \\ _Andrew Howroyd_, Jul 15 2018

%Y Cf. A000041, A051908, A058360, A316855, A316856, A316857.

%K nonn

%O 1,4

%A _Gus Wiseman_, Jul 14 2018

%E a(51)-a(77) from _Giovanni Resta_, Jul 15 2018