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A321271 Number of connected factorizations of n into positive integers > 1 with z-density -1. 2
0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 7, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 7, 5, 1, 1, 3, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

These are z-trees (A303837, A305081, A305253, A321279) where we relax the requirement of pairwise indivisibility.

Given a finite multiset S of positive integers greater than 1, let G(S) be the simple labeled graph with vertices the distinct elements of S and with edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. Then S is said to be connected if G(S) is a connected graph.

The z-density of a factorization S is defined to be Sum_{s in S} (omega(s) - 1) - omega(n), where omega = A001221 and n is the product of S.

LINKS

Table of n, a(n) for n=1..87.

EXAMPLE

The a(72) = 8 factorizations are (2*2*3*6), (2*2*18), (2*3*12), (2*36), (3*4*6), (3*24), (4*18), (72). Missing from this list but still connected are (2*6*6),(6*12).

MATHEMATICA

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c=={}, s, zsm[Union[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];

zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[Times@@s];

Table[Length[Select[facs[n], And[zensity[#]==-1, Length[zsm[#]]==1]&]], {n, 100}]

CROSSREFS

Cf. A001055, A001221, A030019, A286518, A303837, A304118, A304382, A305052, A305081, A305193, A305253, A319786, A321229, A321253.

Sequence in context: A328855 A327658 A319786 * A305193 A038538 A293515

Adjacent sequences:  A321268 A321269 A321270 * A321272 A321273 A321274

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 01 2018

STATUS

approved

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Last modified October 29 19:16 EDT 2020. Contains 338067 sequences. (Running on oeis4.)