A350139 Number of non-weakly alternating ordered factorizations of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 12, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 20, 0, 0, 0, 0, 0, 2, 0, 10, 0, 0, 0, 12, 0

Offset

1,24

Comments

The first odd term is a(180) = 69, which has, for example, the non-weakly alternating ordered factorization 2*3*5*3*2.

An ordered factorization of n is a finite sequence of positive integers > 1 with product n. Ordered factorizations are counted by A074206.

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

Links

Table of n, a(n) for n=1..85.

Formula

a(2^n) = A349053(n).

Example

The a(n) ordered factorizations for n = 24, 36, 48, 60:
  (2*3*4)  (2*3*6)    (2*3*8)    (2*5*6)
  (4*3*2)  (6*3*2)    (2*4*6)    (3*4*5)
           (2*3*3*2)  (6*4*2)    (5*4*3)
           (3*2*2*3)  (8*3*2)    (6*5*2)
                      (2*2*3*4)  (10*3*2)
                      (2*3*4*2)  (2*3*10)
                      (2*4*3*2)  (2*2*3*5)
                      (3*2*2*4)  (2*3*5*2)
                      (4*2*2*3)  (2*5*3*2)
                      (4*3*2*2)  (3*2*2*5)
                                 (5*2*2*3)
                                 (5*3*2*2)

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
whkQ[y_]:=And@@Table[If[EvenQ[m], y[[m]]<=y[[m+1]], y[[m]]>=y[[m+1]]], {m, 1, Length[y]-1}];
Table[Length[Select[Join@@Permutations/@facs[n], !whkQ[#]&&!whkQ[-#]&]], {n, 100}]

Crossrefs

Positions of nonzero terms are A122181.

The strong version for compositions is A345192, ranked by A345168.

The strong case is A348613, complement A348610.

The version for compositions is A349053, complement A349052.

As compositions with ones allowed these are ranked by A349057.

The complement is counted by A349059.

A001055 counts factorizations, strict A045778, ordered A074206.

A001250 counts alternating permutations, complement A348615.

A025047 counts weakly alternating compositions, ranked by A345167.

A335434 counts separable factorizations, complement A333487.

A345164 counts alternating perms of prime factors, with twins A344606.

A345170 counts partitions with an alternating permutation.

A348379 counts factorizations w/ alternating perm, complement A348380.

A348611 counts anti-run ordered factorizations, complement A348616.

A349060 counts weakly alternating partitions, complement A349061.

Cf. A003242, A138364, A339846, A339890, A344604, A345194, A347050, A347438, A347463, A347706, A349054.

Sequence in context: A000086 A363858 A045838 * A293814 A045837 A319687

Adjacent sequences: A350136 A350137 A350138 * A350140 A350141 A350142

Keyword

nonn

Author

Gus Wiseman, Dec 24 2021

Status

approved