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%I #5 Oct 20 2019 22:05:16
%S 0,1,1,1,1,2,1,1,1,2,1,2,1,2,2,1,1,2,1,2,1,2,1,2,1,2,1,2,1,3,1,1,2,2,
%T 2,2,1,2,1,2,1,2,1,2,2,2,1,2,1,2,2,2,1,2,2,2,1,2,1,3,1,2,1,1,1,3,1,2,
%U 2,3,1,2,1,2,2,2,2,2,1,2,1,2,1,2,2,2,1
%N Number of distinct connected components of the multiset of multisets with MM-number n.
%C For n > 1, the first appearance of n is 2 * A080696(n - 1).
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
%F If n is even, a(n) = A305079(n) - A007814(n) + 1; otherwise, a(n) = A305079(n).
%e The multiset of multisets with MM-number 1508 is {{},{},{1,2},{1,3}}, which has the 3 connected components {{}}, {{}}, and {{1,2},{1,3}}, two of which are distinct, so a(1508) = 2.
%e The multiset of multisets with MM-number 12818 is {{},{1,2},{4},{1,3}}, which has the 3 connected components {{}}, {{1,2},{1,3}}, and {{4}}, so a(12818) = 3.
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
%t Table[Length[Union[zsm[primeMS[n]]]],{n,100}]
%Y Positions of 0's and 1's are A305078 together with all powers of 2.
%Y Connected numbers are A305078.
%Y Connected components are A305079.
%Y Cf. A007814, A056239, A112798, A286518, A302242, A304714, A304716, A322389, A327076, A328513.
%K nonn
%O 1,6
%A _Gus Wiseman_, Oct 20 2019
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