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

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, 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, 2, 3, 1, 3, 1, 2, 3, 2, 2, 4, 1, 2, 4, 2, 1, 4, 2, 2, 3

Offset: 1

Author

Gus Wiseman, Dec 03 2018

Keywords

• nonn

Comments

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).

Examples

			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}}.
     3: {{1}}
     5: {{2}}
     7: {{1,1}}
    13: {{1,2}}
    21: {{1},{1,1}}
    39: {{1},{1,2}}
    65: {{2},{1,2}}
    91: {{1,1},{1,2}}
   195: {{1},{2},{1,2}}
   273: {{1},{1,1},{1,2}}
   455: {{2},{1,1},{1,2}}
  1365: {{1},{2},{1,1},{1,2}}
		

Crossrefs

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

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]]]]]]];
Table[Length[Select[Union[Subsets[primeMS[n]]],Length[zsm[#]]==1&]],{n,50}]