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A322400
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Heinz numbers of integer partitions with vertex-connectivity 1.
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2
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3, 5, 7, 9, 11, 17, 19, 21, 23, 25, 27, 31, 41, 49, 53, 57, 59, 63, 67, 81, 83, 97, 103, 109, 115, 121, 125, 127, 131, 133, 147, 157, 159, 171, 179, 189, 191, 211, 227, 241, 243, 277, 283, 289, 311, 331, 343, 353, 361, 367, 371, 377, 393, 399, 401, 419, 431
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OFFSET
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1,1
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COMMENTS
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The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
The vertex-connectivity of an integer partition is the minimum number of primes that must be divided out (and any parts then equal to 1 removed) so that the prime factorizations of the remaining parts form a disconnected (or empty) hypergraph.
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LINKS
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EXAMPLE
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The sequence of all integer partitions with vertex-connectivity 1 begins: (2), (3), (4), (2,2), (5), (7), (8), (4,2), (9), (3,3), (2,2,2), (11), (13), (4,4), (16), (8,2), (17), (4,2,2), (19), (2,2,2,2), (23), (25), (27), (29), (9,3), (5,5), (3,3,3), (31), (32), (8,4), (4,4,2), (37), (16,2), (8,2,2), (41), (4,2,2,2), (43).
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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]]]]]]]]];
vertConn[y_]:=If[Length[csm[primeMS/@y]]!=1, 0, Min@@Length/@Select[Subsets[Union@@primeMS/@y], Function[del, Length[csm[DeleteCases[DeleteCases[primeMS/@y, Alternatives@@del, {2}], {}]]]!=1]]];
Select[Range[100], vertConn[primeMS[#]]==1&]
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CROSSREFS
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Cf. A003963, A013922, A056239, A112798, A302242, A304716, A305078, A305079, A322387, A322388, A322389, A322390, A322390, A322394.
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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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