A348609 Numbers with a separable factorization without an alternating permutation.

216, 270, 324, 378, 432, 486, 540, 594, 640, 648, 702, 756, 768, 810, 864, 896, 918, 960, 972, 1024, 1026, 1080, 1134, 1152, 1188, 1242, 1280, 1296, 1344, 1350, 1404, 1408, 1458, 1500, 1512, 1536, 1566, 1620, 1664, 1674, 1728, 1750, 1782, 1792, 1836, 1890

Offset

1,1

Comments

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

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts. Alternatively, a multiset is separable if its greatest multiplicity is greater than the sum of the remaining multiplicities plus one.

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). Alternating permutations of multisets are a generalization of alternating or up-down permutations of sets.

Note that 216 has separable prime factorization (2*2*2*3*3*3) with an alternating permutation, but the separable factorization (2*3*3*3*4) is has no alternating permutation. See also A345173.

Links

Table of n, a(n) for n=1..46.

Example

The terms and their prime factorizations begin:
  216 = 2*2*2*3*3*3
  270 = 2*3*3*3*5
  324 = 2*2*3*3*3*3
  378 = 2*3*3*3*7
  432 = 2*2*2*2*3*3*3
  486 = 2*3*3*3*3*3
  540 = 2*2*3*3*3*5
  594 = 2*3*3*3*11
  640 = 2*2*2*2*2*2*2*5
  648 = 2*2*2*3*3*3*3
  702 = 2*3*3*3*13
  756 = 2*2*3*3*3*7
  768 = 2*2*2*2*2*2*2*2*3
  810 = 2*3*3*3*3*5
  864 = 2*2*2*2*2*3*3*3

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
sepQ[m_]:=Select[Permutations[m], !MatchQ[#, {___, x_, x_, ___}]&]!={};
wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Select[Range[1000], Function[n, Select[facs[n], sepQ[#]&&Select[Permutations[#], wigQ]=={}&]!={}]]

Crossrefs

Partitions of this type are counted by A345166, ranked by A345173 (a superset).

Compositions of this type are counted by A345195, ranked by A345169.

A001055 counts factorizations, strict A045778, ordered A074206.

A001250 counts alternating permutations, complement A348615.

A025047 counts alternating compositions, complement A345192, ranked by A345167.

A335434 counts separable factorizations, with twins A348383, complement A333487.

A339846 counts even-length factorizations.

A339890 counts odd-length factorizations.

A345165 counts partitions w/o an alternating permutation, complement A345170.

A347438 counts factorizations with alternating product 1, additive A119620.

A348379 counts factorizations w/ an alternating permutation, complement A348380.

A348610 counts alternating ordered factorizations, complement A348613.

Cf. A001248, A003242, A038548, A325534, A336103, A344614, A345162, A347050, A347437, A347706, A348377, A348381, A348382.

Sequence in context: A230400 A001239 A184088 * A205191 A204650 A115430

Adjacent sequences: A348606 A348607 A348608 * A348610 A348611 A348612

Keyword

nonn

Author

Gus Wiseman, Oct 30 2021

Status

approved