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A322368
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Heinz numbers of disconnected integer partitions.
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2
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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
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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).
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MATHEMATICA
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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&]
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CROSSREFS
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Cf. A054921, A218970, A286518, A290103, A304714, A304716, A305078, A305079, A322306, A322307, A322338, A322367, A322369.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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