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A322394
Heinz numbers of integer partitions with edge-connectivity 1.
5
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 195, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269
OFFSET
1,1
COMMENTS
The first nonprime term is 195, which is the Heinz number of (6,3,2).
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
An integer partition has edge-connectivity 1 if the prime factorizations of the parts form a connected hypergraph that can be disconnected (or made empty) by removing a single part.
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]]]]]]]]];
edgeConn[y_]:=If[Length[csm[primeMS/@y]]!=1, 0, Length[y]-Max@@Length/@Select[Union[Subsets[y]], Length[csm[primeMS/@#]]!=1&]];
Select[Range[100], edgeConn[primeMS[#]]==1&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 06 2018
STATUS
approved