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A305504
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Heinz numbers of integer partitions whose distinct parts plus 1 are connected.
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1
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1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 22, 23, 25, 27, 29, 31, 32, 33, 34, 37, 40, 41, 43, 44, 46, 47, 49, 50, 53, 55, 57, 59, 61, 62, 64, 66, 67, 68, 71, 73, 79, 80, 81, 82, 83, 85, 88, 89, 92, 93, 94, 97, 99, 100, 101, 103, 107, 109, 110, 113, 115
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OFFSET
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1,2
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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).
Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A partition y is said to be connected if G(U(y + 1)) is a connected graph, where U(y + 1) is the set of distinct successors of the parts of y.
This is intended to be a cleaner form of A305078, where the treatment of empty multisets is arbitrary.
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LINKS
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EXAMPLE
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The sequence of entries together with the corresponding twice-prime-factored multiset partitions (see A275024) begins:
1: {}
2: {{1}}
3: {{2}}
4: {{1},{1}}
5: {{1,1}}
7: {{3}}
8: {{1},{1},{1}}
9: {{2},{2}}
10: {{1},{1,1}}
11: {{1,2}}
13: {{4}}
16: {{1},{1},{1},{1}}
17: {{1,1,1}}
19: {{2,2}}
20: {{1},{1},{1,1}}
22: {{1},{1,2}}
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MATHEMATICA
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primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c=={}, s, zsm[Union[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];
Select[Range[300], Length[zsm[primeMS[#]+1]]<=1&]
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CROSSREFS
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Cf. A001221, A048143, A056239, A112798, A275024, A286518, A290103, A302242, A304714, A304716, A305052, A305078, A305079, A305501.
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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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