A327392 Irregular triangle read by rows giving the connected components of the prime indices of n.

1, 2, 1, 1, 3, 1, 2, 4, 1, 1, 1, 2, 1, 3, 5, 1, 1, 2, 6, 1, 4, 2, 3, 1, 1, 1, 1, 7, 1, 2, 8, 1, 1, 3, 4, 1, 5, 9, 1, 1, 1, 2, 3, 1, 6, 2, 1, 1, 4, 10, 1, 2, 3, 11, 1, 1, 1, 1, 1, 2, 5, 1, 7, 3, 4, 1, 1, 2, 12, 1, 8, 6, 1, 1, 1, 3, 13, 1, 4, 14, 1, 1, 5, 2, 3

Offset: 1

Author

Gus Wiseman, Oct 03 2019

Keywords

• nonn
• tabf

Comments

First differs from A112798 at a(13) = 1, A112798(13) = 2.

The terms of each row are pairwise coprime.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

A number n with prime factorization n = prime(m_1)^s_1 * ... * prime(m_k)^s_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

			Triangle begins:
  {}
  1
  2
  1 1
  3
  1 2
  4
  1 1 1
  2
  1 3
  5
  1 1 2
  6
  1 4
  2 3
  1 1 1 1
  7
  1 2
  8
  1 1 3
  4
  1 5
  9
  1 1 1 2
  3
  1 6
  2
  1 1 4
		

Crossrefs

Row lengths are A305079.

Cf. A000005, A056239, A112798, A218970, A304716, A302242, A305078, A327076.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],GCD@@s[[#]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Table[zsm[primeMS[n]],{n,30}]