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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
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
Sequence in context: A328855 A327658 A319786 * A305193 A038538 A293515
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 01 2018
STATUS
approved

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Last modified March 28 07:20 EDT 2024. Contains 371235 sequences. (Running on oeis4.)