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

1,2

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.

Links

Table of n, a(n) for n=1..59.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

Formula

Intersection of A005117 and A305078.

Example

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}}

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&]

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.

Sequence in context: A230606 A117289 A339817 * A318719 A106317 A246281

Adjacent sequences: A328510 A328511 A328512 * A328514 A328515 A328516

Keyword

nonn

Author

Gus Wiseman, Oct 20 2019

Status

approved