

A321271


Number of connected factorizations of n into positive integers > 1 with zdensity 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
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OFFSET

1,4


COMMENTS

These are ztrees (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 4cycle. Then S is said to be connected if G(S) is a connected graph.
The zdensity 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



