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A327392
Irregular triangle read by rows giving the connected components of the prime indices of n.
2
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.
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
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Oct 03 2019
STATUS
approved