OFFSET
1,60
COMMENTS
A factorization into factors > 1 is unsortable if there is no permutation (c_1,...,c_k) of the factors such that the maximum prime factor of c_i is at most the minimum prime factor of c_{i+1}. For example, the factorization (6*8*27) is sortable because the permutation (8,6,27) satisfies the condition.
EXAMPLE
The a(180) = 10 unsortable factorizations:
(2*3*3*10) (5*6*6) (3*60)
(2*3*30) (6*30)
(2*9*10) (9*20)
(3*3*20) (10*18)
(3*6*10)
Missing from this list are:
(2*2*3*3*5) (2*2*5*9) (4*5*9) (2*90) (180)
(2*3*5*6) (2*2*45) (4*45)
(3*3*4*5) (2*5*18) (5*36)
(2*2*3*15) (2*6*15) (12*15)
(3*4*15)
(3*5*12)
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
lexsort[f_, c_]:=OrderedQ[PadRight[{f, c}]];
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Select[facs[n], !OrderedQ[Join@@Sort[primeMS/@#, lexsort]]&]], {n, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 24 2019
STATUS
approved