A345171 Numbers whose multiset of prime factors has no alternating permutation.

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 80, 81, 88, 96, 104, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 270, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336, 343, 344, 351

Offset: 1

Author

Gus Wiseman, Jun 13 2021

Keywords

• nonn

Comments

First differs from A335448 in having 270.

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 has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

Also Heinz numbers of integer partitions without a wiggly permutation, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   16: {1,1,1,1}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   32: {1,1,1,1,1}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   49: {4,4}
   54: {1,2,2,2}
   56: {1,1,1,4}
   64: {1,1,1,1,1,1}
   80: {1,1,1,1,3}
   81: {2,2,2,2}
   88: {1,1,1,5}
   96: {1,1,1,1,1,2}
		

Crossrefs

Removing squares of primes A001248 gives A344653, counted by A344654.

A superset of A335448, which is counted by A325535.

Positions of 0's in A345164.

The partitions with these Heinz numbers are counted by A345165.

The complement is A345172, counted by A345170.

The separable case is A345173, counted by A345166.

A001250 counts alternating permutations, complement A348615.

A003242 counts anti-run compositions, complement A261983.

A025047 counts alternating or wiggly compositions, directed A025048, A025049.

A325534 counts separable partitions, ranked by A335433.

A344606 counts alternating permutations of prime indices with twins.

A344742 ranks twins and partitions with an alternating permutation.

A345192 counts non-alternating compositions.

Cf. A001222, A071321, A071322, A316523, A316524, A335126, A344604, A344616, A344652, A344740, A345163, A345168, A345193, A345195, A348380, A348609.

Programs

Mathematica

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Select[Range[100],Select[Permutations[Flatten[ ConstantArray@@@FactorInteger[#]]],wigQ]=={}&]