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

1,2

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.

Links

Table of n, a(n) for n=1..85.

Example

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

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}]

Crossrefs

Row lengths are A305079.

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

Sequence in context: A134521 A131375 A358169 * A112798 A187846 A181087

Adjacent sequences: A327389 A327390 A327391 * A327393 A327394 A327395

Keyword

nonn,tabf

Author

Gus Wiseman, Oct 03 2019

Status

approved