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Number of connected divisors of n. Number of connected submultisets of the n-th multiset multisystem (A302242).
6

%I #6 Dec 03 2018 21:38:20

%S 0,1,1,1,1,2,1,1,2,2,1,2,1,2,2,1,1,3,1,2,3,2,1,2,2,2,3,2,1,3,1,1,2,2,

%T 2,3,1,2,3,2,1,4,1,2,3,2,1,2,2,3,2,2,1,4,2,2,3,2,1,3,1,2,5,1,3,3,1,2,

%U 2,3,1,3,1,2,3,2,2,4,1,2,4,2,1,4,2,2,3

%N Number of connected divisors of n. Number of connected submultisets of the n-th multiset multisystem (A302242).

%C A prime index of n is a number m such that prime(m) divides n. A positive integer is connected if its prime indices are connected (see A305078).

%e The a(1365) = 12 divisors are 3, 5, 7, 13, 21, 39, 65, 91, 195, 273, 455, 1365. These correspond to the following connected submultisets of {{1},{2},{1,1},{1,2}}.

%e 3: {{1}}

%e 5: {{2}}

%e 7: {{1,1}}

%e 13: {{1,2}}

%e 21: {{1},{1,1}}

%e 39: {{1},{1,2}}

%e 65: {{2},{1,2}}

%e 91: {{1,1},{1,2}}

%e 195: {{1},{2},{1,2}}

%e 273: {{1},{1,1},{1,2}}

%e 455: {{2},{1,1},{1,2}}

%e 1365: {{1},{2},{1,1},{1,2}}

%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[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];

%t Table[Length[Select[Union[Subsets[primeMS[n]]],Length[zsm[#]]==1&]],{n,50}]

%Y Cf. A003963, A054921, A112798, A286518, A290103, A301957, A302242, A304714, A304716, A305078, A316556, A322307.

%K nonn

%O 1,6

%A _Gus Wiseman_, Dec 03 2018