cp's OEIS Frontend

A322306 Number of connected divisors of n. Number of connected submultisets of the n-th multiset multisystem (A302242).

%I A322306 #6 Dec 03 2018 21:38:20
%S A322306 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 A322306 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 A322306 2,3,1,3,1,2,3,2,2,4,1,2,4,2,1,4,2,2,3
%N A322306 Number of connected divisors of n. Number of connected submultisets of the n-th multiset multisystem (A302242).
%C A322306 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 A322306 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 A322306      3: {{1}}
%e A322306      5: {{2}}
%e A322306      7: {{1,1}}
%e A322306     13: {{1,2}}
%e A322306     21: {{1},{1,1}}
%e A322306     39: {{1},{1,2}}
%e A322306     65: {{2},{1,2}}
%e A322306     91: {{1,1},{1,2}}
%e A322306    195: {{1},{2},{1,2}}
%e A322306    273: {{1},{1,1},{1,2}}
%e A322306    455: {{2},{1,1},{1,2}}
%e A322306   1365: {{1},{2},{1,1},{1,2}}
%t A322306 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A322306 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 A322306 Table[Length[Select[Union[Subsets[primeMS[n]]],Length[zsm[#]]==1&]],{n,50}]
%Y A322306 Cf. A003963, A054921, A112798, A286518, A290103, A301957, A302242, A304714, A304716, A305078, A316556, A322307.
%K A322306 nonn
%O A322306 1,6
%A A322306 _Gus Wiseman_, Dec 03 2018