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%I #4 Oct 04 2019 23:29:51
%S 1,2,1,1,3,1,2,4,1,1,1,2,1,3,5,1,1,2,6,1,4,2,3,1,1,1,1,7,1,2,8,1,1,3,
%T 4,1,5,9,1,1,1,2,3,1,6,2,1,1,4,10,1,2,3,11,1,1,1,1,1,2,5,1,7,3,4,1,1,
%U 2,12,1,8,6,1,1,1,3,13,1,4,14,1,1,5,2,3
%N Irregular triangle read by rows giving the connected components of the prime indices of n.
%C First differs from A112798 at a(13) = 1, A112798(13) = 2.
%C The terms of each row are pairwise coprime.
%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.
%C A number n with prime factorization n = prime(m_1)^s_1 * ... * prime(m_k)^s_k is connected if the simple labeled graph with vertex set {m_1,...,m_k} and edges between any two vertices with a common divisor greater than 1 is connected. Connected numbers are listed in A305078.
%e Triangle begins:
%e {}
%e 1
%e 2
%e 1 1
%e 3
%e 1 2
%e 4
%e 1 1 1
%e 2
%e 1 3
%e 5
%e 1 1 2
%e 6
%e 1 4
%e 2 3
%e 1 1 1 1
%e 7
%e 1 2
%e 8
%e 1 1 3
%e 4
%e 1 5
%e 9
%e 1 1 1 2
%e 3
%e 1 6
%e 2
%e 1 1 4
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t zsm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],GCD@@s[[#]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
%t Table[zsm[primeMS[n]],{n,30}]
%Y Row lengths are A305079.
%Y Cf. A000005, A056239, A112798, A218970, A304716, A302242, A305078, A327076.
%K nonn,tabf
%O 1,2
%A _Gus Wiseman_, Oct 03 2019
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