A328513 Connected squarefree numbers.

1, 2, 3, 5, 7, 11, 13, 17, 19, 21, 23, 29, 31, 37, 39, 41, 43, 47, 53, 57, 59, 61, 65, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 127, 129, 131, 133, 137, 139, 149, 151, 157, 159, 163, 167, 173, 179, 181, 183, 185, 191, 193, 195

Offset: 1

Author

Gus Wiseman, Oct 20 2019

Keywords

• nonn

Comments

First differs from A318718 and A318719 in having 195 = prime(2) * prime(3) * prime(6).

A squarefree number with prime factorization prime(m_1) * ... * prime(m_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.

Examples

			The sequence of all connected sets of multisets together with their MM-numbers (A302242) begins:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
  11: {{3}}
  13: {{1,2}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  23: {{2,2}}
  29: {{1,3}}
  31: {{5}}
  37: {{1,1,2}}
  39: {{1},{1,2}}
  41: {{6}}
  43: {{1,4}}
  47: {{2,3}}
  53: {{1,1,1,1}}
  57: {{1},{1,1,1}}
		

Links

• Gus Wiseman, Sequences counting and encoding certain classes of multisets

Crossrefs

A subset of A005117.

These are Heinz numbers of the partitions counted by A304714.

The maximum connected squarefree divisor of n is A327398(n).

Cf. A056239, A112798, A286518, A302242, A304716, A305078, A305079, A327076, A328514.

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[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Select[Range[100],SquareFreeQ[#]&&Length[zsm[primeMS[#]]]<=1&]

Formula

Intersection of A005117 and A305078.