%I #13 Jun 13 2021 10:21:15

%S 8,16,24,27,32,40,48,54,56,64,80,81,88,96,104,112,125,128,135,136,144,

%T 152,160,162,176,184,189,192,208,224,232,240,243,248,250,256,270,272,

%U 288,296,297,304,320,324,328,336,343,344,351,352,368,375,376,378,384

%N Every permutation of the prime factors of n has a consecutive monotone triple, i.e., a triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

%C Differs from A335448 in lacking squares and having 270 etc.

%C First differs from A345193 in having 270.

%C Such a permutation is characterized by being neither a twin (x,x) nor wiggly (A025047, A345192). A sequence is wiggly if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no wiggly permutations, even though it has anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

%F Complement of A001248 in A345171.

%e The sequence of terms together with their prime indices begins:

%e 8: {1,1,1}

%e 16: {1,1,1,1}

%e 24: {1,1,1,2}

%e 27: {2,2,2}

%e 32: {1,1,1,1,1}

%e 40: {1,1,1,3}

%e 48: {1,1,1,1,2}

%e 54: {1,2,2,2}

%e 56: {1,1,1,4}

%e 64: {1,1,1,1,1,1}

%e 80: {1,1,1,1,3}

%e 81: {2,2,2,2}

%e 88: {1,1,1,5}

%e 96: {1,1,1,1,1,2}

%e For example, 36 has prime indices (1,1,2,2), which has the two wiggly permutations (1,2,1,2) and (2,1,2,1), so 36 is not in the sequence.

%t Select[Range[100],Select[Permutations[Flatten[ConstantArray@@@FactorInteger[#]]],!MatchQ[#,{___,x_,y_,z_,___}/;x<=y<=z||x>=y>=z]&]=={}&]

%Y A superset of A335448, counted by A325535.

%Y Positions of 0's in A344606.

%Y These partitions are counted by A344654.

%Y The complement is A344742, counted by A344740.

%Y The separable case is A345173, counted by A345166.

%Y A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.

%Y A001250 counts wiggly permutations.

%Y A003242 counts anti-run compositions.

%Y A025047 counts wiggly compositions (ascend: A025048, descend: A025049).

%Y A325534 counts separable partitions, ranked by A335433.

%Y A344604 counts wiggly compositions with twins.

%Y A345164 counts wiggly permutations of prime indices.

%Y A345165 counts partitions without a wiggly permutation, ranked by A345171.

%Y A345170 counts partitions with a wiggly permutation, ranked by A345172.

%Y A345192 counts non-wiggly compositions.

%Y Cf. A001222, A071321, A071322, A316523, A316524, A335126, A344614, A344615, A344616, A344652, A345163, A345168, A345193.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jun 12 2021