A322368 Heinz numbers of disconnected integer partitions.

1, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104

Offset: 1

Author

Gus Wiseman, Dec 04 2018

Keywords

• nonn

Comments

Differs from A289509 in having 1 and lacking 2, 195, 455, 555, 585...

Also positions of entries > 1 in A305079.

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

An integer partition is connected if the prime factorizations of its parts form a connected hypergraph. It is disconnected if it can be separated into two or more integer partitions with relatively prime products. For example, the integer partition (654321) has three connected components: (6432)(5)(1).

Examples

			The sequence of all disconnected integer partitions begins: (11), (21), (111), (31), (211), (41), (32), (1111), (221), (311), (51), (2111), (61), (411), (321), (11111), (52), (71), (43), (2211), (81), (3111), (421), (511), (322), (91), (21111), (331), (72), (611), (2221), (53), (4111).
		

Crossrefs

Cf. A054921, A218970, A286518, A290103, A304714, A304716, A305078, A305079, A322306, A322307, A322338, A322367, A322369.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Select[Range[200],Length[csm[primeMS/@primeMS[#]]]>1&]