A350353 Numbers whose multiset of prime factors has a permutation that is not weakly alternating.

30, 36, 42, 60, 66, 70, 72, 78, 84, 90, 100, 102, 105, 108, 110, 114, 120, 126, 130, 132, 138, 140, 144, 150, 154, 156, 165, 168, 170, 174, 180, 182, 186, 190, 195, 196, 198, 200, 204, 210, 216, 220, 222, 225, 228, 230, 231, 234, 238, 240, 246, 252, 255, 258

Offset: 1

Author

Gus Wiseman, Jan 13 2022

Keywords

• nonn

Comments

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

Examples

			The terms together with a (generally not unique) non-weakly alternating permutation of each multiset of prime indices begin:
   30 : (1,2,3)       100 : (1,3,3,1)
   36 : (1,2,2,1)     102 : (1,2,7)
   42 : (1,2,4)       105 : (2,3,4)
   60 : (1,1,2,3)     108 : (1,2,2,1,2)
   66 : (1,2,5)       110 : (1,3,5)
   70 : (1,3,4)       114 : (1,2,8)
   72 : (1,1,2,2,1)   120 : (1,1,1,2,3)
   78 : (1,2,6)       126 : (1,2,4,2)
   84 : (1,1,2,4)     130 : (1,3,6)
   90 : (1,2,3,2)     132 : (1,1,2,5)
		

Crossrefs

The strong version is A289553, complement A167171.

These are the positions of nonzero terms in A349797.

Below, WA = "weakly alternating":

- WA compositions are counted by A349052/A129852/A129853.

- Non-WA compositions are counted by A349053, ranked by A349057.

- WA permutations of prime factors = A349056, complement A349797.

- WA patterns are counted by A349058, complement A350138.

- WA ordered factorizations are counted by A349059, complement A350139.

- WA partitions are counted by A349060, complement A349061.

A001250 counts alternating permutations, complement A348615.

A008480 counts permutations of prime factors.

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

A056239 adds up prime indices, row sums of A112798 (row lengths A001222).

A071321 gives the alternating sum of prime factors, reverse A071322.

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

A345164 = alternating permutations of prime factors, complement A350251.

Cf. A003242, A335433, A335448, A344652, A344653, A345171, A345172, A345173, A348379, A348613, A349798, A350252, A349800.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Select[Range[100],Select[Permutations[primeMS[#]],!whkQ[#]&&!whkQ[-#]&]!={}&]