A348381 Number of inseparable factorizations of n that are not a twin (x*x).

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0

Offset

1,32

Comments

First differs from A347706 at a(216) = 3, A347706(216) = 4.

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

A multiset is inseparable if it has no permutation that is an anti-run, meaning there are always adjacent equal parts. Alternatively, a multiset is inseparable if its maximal multiplicity is at most one plus the sum of its remaining multiplicities.

Links

Table of n, a(n) for n=1..87.

Formula

a(n > 1) = A333487(n) - A010052(n).

a(2^n) = A325535(n) - 1 for odd n, otherwise A325535(n).

Example

The a(n) factorizations for n = 96, 192, 384, 576:
  2*2*2*12      3*4*4*4         4*4*4*6           4*4*4*9
  2*2*2*2*6     2*2*2*24        2*2*2*48          2*2*2*72
  2*2*2*2*2*3   2*2*2*2*12      2*2*2*2*24        2*2*2*2*36
                2*2*2*2*2*6     2*2*2*2*3*8       2*2*2*2*4*9
                2*2*2*2*3*4     2*2*2*2*4*6       2*2*2*2*6*6
                2*2*2*2*2*2*3   2*2*2*2*2*12      2*2*2*2*2*18
                                2*2*2*2*2*2*6     2*2*2*2*3*12
                                2*2*2*2*2*3*4     2*2*2*2*2*2*9
                                2*2*2*2*2*2*2*3   2*2*2*2*2*3*6
                                                  2*2*2*2*2*2*3*3

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], !MatchQ[#, {x_, x_}]&&Select[Permutations[#], !MatchQ[#, {___, x_, x_, ___}]&]=={}&]], {n, 100}]

Crossrefs

Positions of nonzero terms are A046099.

Partitions not of this type are counted by A325534 - A000035.

Partitions of this type are counted by A325535 - A000035.

Allowing twins gives A333487.

The case without an alternating permutation is A347706, with twins A348380.

The complement is counted by A348383, without twins A335434.

A001055 counts factorizations, strict A045778, ordered A074206.

A001250 counts alternating permutations of sets.

A008480 counts permutations of prime indices, strict A335489.

A025047 counts alternating or wiggly compositions.

A335452 counts anti-run permutations of prime indices, complement A336107.

A339846 counts even-length factorizations.

A339890 counts odd-length factorizations.

A344654 counts non-twin partitions without an alternating permutation.

A348382 counts non-anti-run compositions that are not a twin.

A348611 counts anti-run ordered factorizations.

Cf. A038548, A336102, A344614, A344653, A344740, A347050, A347437, A347438, A347456, A348379, A348609.

Sequence in context: A366123 A290081 A347706 * A010103 A086078 A353462

Adjacent sequences: A348378 A348379 A348380 * A348382 A348383 A348384

Keyword

nonn

Author

Gus Wiseman, Oct 30 2021

Status

approved