cp's OEIS Frontend

A326291 Number of unsortable factorizations of n.

%I A326291 #6 Jun 25 2019 10:10:48
%S A326291 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
%T A326291 0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,
%U A326291 0,1,0,2,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0
%N A326291 Number of unsortable factorizations of n.
%C A326291 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.
%e A326291 The a(180) = 10 unsortable factorizations:
%e A326291   (2*3*3*10)  (5*6*6)   (3*60)
%e A326291               (2*3*30)  (6*30)
%e A326291               (2*9*10)  (9*20)
%e A326291               (3*3*20)  (10*18)
%e A326291               (3*6*10)
%e A326291 Missing from this list are:
%e A326291   (2*2*3*3*5)  (2*2*5*9)   (4*5*9)   (2*90)   (180)
%e A326291                (2*3*5*6)   (2*2*45)  (4*45)
%e A326291                (3*3*4*5)   (2*5*18)  (5*36)
%e A326291                (2*2*3*15)  (2*6*15)  (12*15)
%e A326291                            (3*4*15)
%e A326291                            (3*5*12)
%t A326291 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A326291 lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
%t A326291 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A326291 Table[Length[Select[facs[n],!OrderedQ[Join@@Sort[primeMS/@#,lexsort]]&]],{n,100}]
%Y A326291 Unsortable set partitions are A058681.
%Y A326291 Unsortable normal multiset partitions are A326211.
%Y A326291 MM-numbers of unsortable multiset partitions are A326258.
%Y A326291 Cf. A000108, A001055, A001519, A016098, A112798, A302242, A324170, A324171.
%Y A326291 Cf. A326209, A326212, A326243, A326256, A326257.
%K A326291 nonn
%O A326291 1,60
%A A326291 _Gus Wiseman_, Jun 24 2019