cp's OEIS Frontend

A322338 Edge-connectivity of the integer partition with Heinz number n.

%I A322338 #5 Dec 05 2018 07:58:26
%S A322338 0,1,1,0,1,0,1,0,2,0,1,0,1,0,0,0,1,0,1,0,2,0,1,0,2,0,3,0,1,0,1,0,0,0,
%T A322338 0,0,1,0,2,0,1,0,1,0,0,0,1,0,2,0,0,0,1,0,0,0,2,0,1,0,1,0,3,0,2,0,1,0,
%U A322338 0,0,1,0,1,0,0,0,0,0,1,0,4,0,1,0,0,0,2
%N A322338 Edge-connectivity of the integer partition with Heinz number n.
%C A322338 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A322338 The edge-connectivity of an integer partition is the minimum number of parts that must be removed so that the prime factorizations of the remaining parts form a disconnected (or empty) hypergraph.
%H A322338 Wikipedia, <a href="https://en.wikipedia.org/wiki/K-edge-connected_graph">k-edge-connected graph</a>
%e A322338 2093 is the Heinz number of (9,6,4), corresponding to the multiset partition {{1,1},{1,2},{2,2}}, which can be made disconnected by removing only the part {1,2}, so a(2093) = 1.
%t A322338 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A322338 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]]]]]]]]];
%t A322338 Table[PrimeOmega[n]-Max@@PrimeOmega/@Select[Divisors[n],Length[csm[primeMS/@primeMS[#]]]!=1&],{n,100}]
%Y A322338 Cf. A001222, A003963, A013922, A054921, A056239, A095983, A112798, A302242, A304714, A304716, A305078, A305079, A322335, A322336, A322337.
%K A322338 nonn
%O A322338 1,9
%A A322338 _Gus Wiseman_, Dec 04 2018