A298676 Number of partitions of n that can be uniquely recovered from their P-graphs.

1, 2, 3, 5, 5, 7, 7, 10, 11, 13, 13, 18, 19, 26, 31, 36, 41, 48, 59, 71, 84, 94, 106, 123, 146, 165, 187, 210, 240, 275, 318, 364, 407, 465, 525, 593, 672, 756, 849, 966, 1080, 1207, 1354, 1530, 1718, 1925, 2135, 2377, 2667, 2997, 3351, 3736, 4141, 4598, 5125

Offset: 1

Author

Bernardo Recamán, Jan 28 2018

Keywords

• nonn

Comments

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.

Examples

			a(1) = 1 because the sole partition of 1 can be recovered from its P-graph, a single vertex.
a(2) = 2 because both partitions of 2 can be recovered from their corresponding P-graphs.
		

Links

• Bernardo Recamán Santos, A unique partition of 200 into 6 parts, Puzzling Stack Exchange, Dec 17 2017.

Programs

Mathematica

pgraph[p_] := With[{v = Range[Length[p]]}, Graph[v, UndirectedEdge @@@ Select[Subsets[v, {2}], !CoprimeQ @@ p[[#]] &]]];
a[n_] := Count[Length /@ Gather[pgraph /@ IntegerPartitions[n], IsomorphicGraphQ], 1];
Array[a, 20]
(* Andrey Zabolotskiy, Jan 30 2018 *)

Extensions

a(23)-a(50) from Freddy Barrera, Jan 29 2018

a(51)-a(55) from Andrey Zabolotskiy, Jan 30 2018