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 A322389 Vertex-connectivity of the integer partition with Heinz number n. 19
 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 2, 0, 2, 0, 1, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,13 COMMENTS 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. LINKS Wikipedia, k-vertex-connected graph EXAMPLE The integer partition (6,4,3) with Heinz number 455 does not become disconnected or empty if 2 is divided out giving (3,3), or if 3 is divided out giving (4,2), but it does become disconnected or empty if both 2 and 3 are divided out giving (); so a(455) = 2. 195 is the Heinz number of (6,3,2), corresponding to the multiset partition {{1},{2},{1,2}}. Removing the vertex 1 gives {{2},{2}}, while removing 2 gives {{1},{1}}. These are both connected, so both vertices must be removed to obtain a disconnected or empty multiset partition; hence a(195) = 2. 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]]]]]]]]]; 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]]]; Array[vertConn@*primeMS, 100] CROSSREFS Cf. A003963, A013922, A056239, A095983, A112798, A302242, A304716, A305078, A305079, A322335, A322336, A322338, A322387, A322388, A322390. Sequence in context: A085491 A321013 A284258 * A336388 A277735 A248911 Adjacent sequences:  A322386 A322387 A322388 * A322390 A322391 A322392 KEYWORD nonn AUTHOR Gus Wiseman, Dec 05 2018 STATUS approved

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Last modified September 23 02:41 EDT 2021. Contains 347609 sequences. (Running on oeis4.)