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%I #41 Jan 30 2018 09:31:30
%S 1,2,3,5,5,7,7,10,11,13,13,18,19,26,31,36,41,48,59,71,84,94,106,123,
%T 146,165,187,210,240,275,318,364,407,465,525,593,672,756,849,966,1080,
%U 1207,1354,1530,1718,1925,2135,2377,2667,2997,3351,3736,4141,4598,5125
%N Number of partitions of n that can be uniquely recovered from their P-graphs.
%C a(n) is the number of partitions of n that can be uniquely recovered from its P-graph, the simple graph whose vertices are the parts of the partition, two of which are joined by an edge if, and only if, they have a common factor greater than 1.
%H Bernardo Recamán Santos, <a href="https://puzzling.stackexchange.com/questions/58136/a-unique-partition-of-200-into-6-parts">A unique partition of 200 into 6 parts</a>, Puzzling Stack Exchange, Dec 17 2017.
%e a(1) = 1 because the sole partition of 1 can be recovered from its P-graph, a single vertex.
%e a(2) = 2 because both partitions of 2 can be recovered from their corresponding P-graphs.
%t pgraph[p_] := With[{v = Range[Length[p]]}, Graph[v, UndirectedEdge @@@ Select[Subsets[v, {2}], !CoprimeQ @@ p[[#]] &]]];
%t a[n_] := Count[Length /@ Gather[pgraph /@ IntegerPartitions[n], IsomorphicGraphQ], 1];
%t Array[a, 20]
%t (* _Andrey Zabolotskiy_, Jan 30 2018 *)
%K nonn
%O 1,2
%A _Bernardo Recamán_, Jan 28 2018
%E a(23)-a(50) from _Freddy Barrera_, Jan 29 2018
%E a(51)-a(55) from _Andrey Zabolotskiy_, Jan 30 2018
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