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A316898 Number of integer partitions of n into relatively prime parts whose reciprocal sum is the reciprocal of an integer.

%I A316898 #30 Jul 22 2018 10:56:39
%S A316898 1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,2,0,0,0,0,0,2,4,1,3,1,1,1,
%T A316898 3,1,8,3,1,1,9,2,11,3,3,3,5,2,7,6,4,7,12,5,14,6,11,12,25,11,27,17,15,
%U A316898 19,25,9,37,20,21,19,31,19,38,33,26,37,38,36,64,39,46,53,63,39,80,63,65,66,94,59,105
%N A316898 Number of integer partitions of n into relatively prime parts whose reciprocal sum is the reciprocal of an integer.
%C A316898 The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.
%C A316898 Records: 1, 2, 4, 8, 9, 11, 12, 14, 25, 27, 37, 38, 64, 80, 94, 105, 119, 154, 184, ..., . - _Robert G. Wilson v_, Jul 18 2018
%H A316898 Robert G. Wilson v, <a href="/A316898/b316898.txt">Table of n, a(n) for n = 1..150</a>
%H A316898 Gus Wiseman, <a href="/A051908/a051908.txt">Sequences counting and ranking integer partitions by their reciprocal sums</a>
%e A316898 The a(37) = 8 partitions: (20,12,5), (15,12,10), (24,8,3,2), (15,10,6,6), (20,5,4,4,4), (15,10,6,3,3), (14,7,7,7,2), (10,10,10,5,2).
%t A316898 f[n_] := Block[{k = 1, s = 0, lmt = 1 + PartitionsP@ n}, While[k < lmt, s += Length[ Select[ IntegerPartitions[n, {k, k}], GCD @@ # == 1 && IntegerQ[1/Sum[1/m, {m, #}]] &]]; k++]; s]; Array[f, 50] (* slightly modified by _Robert G. Wilson v_, Jul 17 2018 *) (* or *)
%t A316898 ric[n_,p_,s_] := If[n==0, If[IntegerQ[1/s] && GCD @@ p == 1, 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, 50] (* _Giovanni Resta_, Jul 18 2018 *)
%Y A316898 Cf. A000837, A002966, A051908, A058360, A100953, A316854, A316888, A316889, A316890, A316891, A316892, A316893, A316904.
%K A316898 nonn
%O A316898 1,22
%A A316898 _Gus Wiseman_, Jul 16 2018
%E A316898 a(51)-a(91) from _Robert G. Wilson v_, Jul 17 2018