OFFSET
1,5
COMMENTS
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
EXAMPLE
The a(190) = 8 factorizations of 585 together with the corresponding multiset partitions of {2,2,3,6}:
(3*3*5*13) {{2},{2},{3},{6}}
(3*3*65) {{2},{2},{3,6}}
(3*5*39) {{2},{3},{2,6}}
(3*195) {{2},{2,3,6}}
(5*9*13) {{3},{2,2},{6}}
(5*117) {{3},{2,2,6}}
(9*65) {{2,2},{3,6}}
(585) {{2,2,3,6}}
MATHEMATICA
nn=100;
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]]]]]]]]];
facsusing[s_, n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facsusing[Select[s, Divisible[n/d, #]&], n/d], Min@@#>=d&]], {d, Select[s, Divisible[n, #]&]}]];
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
y=Select[Range[nn], Length[zsm[primeMS[#]]]==1&];
Table[Length[facsusing[y, n]], {n, y}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 21 2019
STATUS
approved