A344606 Number of alternating permutations of the prime factors of n, counting multiplicity, including twins (x,x).

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

Offset: 1

Author

Gus Wiseman, May 28 2021

Keywords

• nonn

Comments

Differs from A335448 in having a(x^2) = 0 and a(270) = 0.

These are permutations of the prime factors of n, counting multiplicity, with no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z.

The version without twins (x,x) is A345164, which is identical to this sequence except when n is the square of a prime.

Examples

			The permutations for n = 2, 6, 30, 180, 210, 300, 420, 720, 840:
  2   23   253   23253   2537   25253   23275   2323252   232527
      32   325   32325   2735   25352   25273   2325232   232725
           352   32523   3275   32525   25372   2523232   252327
           523   35232   3527   35252   27253             252723
                 52323   3725   52325   27352             272325
                         5273   52523   32527             272523
                         5372           32725             325272
                         5723           35272             327252
                         7253           37252             523272
                         7352           52327             527232
                                        52723             723252
                                        57232             725232
                                        72325
                                        72523
For example, there are no alternating permutations of the prime factors of 270 because the only anti-runs are {3,2,3,5,3} and {3,5,3,2,3}, neither of which is alternating, so a(270) = 0.
		

Crossrefs

The version for permutations is A001250.

The extension to anti-run permutations is A335452.

The version for compositions is A344604.

The version for patterns is A344605.

Positions of zeros are A344653 (counted by A344654).

Not including twins (x,x) gives A345164.

A008480 counts permutations of prime indices (strict: A335489, rank: A333221).

A056239 adds up prime indices, row sums of A112798.

A071321 and A071322 are signed sums of prime factors.

A316523 is a signed sum of prime multiplicities.

A316524 and A344616 are signed sums of prime indices.

A325534 counts separable partitions (ranked by A335433).

A325535 counts inseparable partitions (ranked by A335448).

A344740 counts partitions with an alternating permutation or twin (x,x).

Cf. A000041, A000607, A000961, A001222, A003242, A026424, A028260, A049774, A103919, A181796, A343938, A344652.

Programs

Mathematica

Table[Length[Select[Permutations[Flatten[ConstantArray@@@FactorInteger[n]]],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]],{n,100}]