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A344653 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.

%I A344653 #13 Jun 13 2021 10:21:15
%S A344653 8,16,24,27,32,40,48,54,56,64,80,81,88,96,104,112,125,128,135,136,144,
%T A344653 152,160,162,176,184,189,192,208,224,232,240,243,248,250,256,270,272,
%U A344653 288,296,297,304,320,324,328,336,343,344,351,352,368,375,376,378,384
%N A344653 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 A344653 Differs from A335448 in lacking squares and having 270 etc.
%C A344653 First differs from A345193 in having 270.
%C A344653 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 A344653 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 A344653 Complement of A001248 in A345171.
%e A344653 The sequence of terms together with their prime indices begins:
%e A344653    8: {1,1,1}
%e A344653   16: {1,1,1,1}
%e A344653   24: {1,1,1,2}
%e A344653   27: {2,2,2}
%e A344653   32: {1,1,1,1,1}
%e A344653   40: {1,1,1,3}
%e A344653   48: {1,1,1,1,2}
%e A344653   54: {1,2,2,2}
%e A344653   56: {1,1,1,4}
%e A344653   64: {1,1,1,1,1,1}
%e A344653   80: {1,1,1,1,3}
%e A344653   81: {2,2,2,2}
%e A344653   88: {1,1,1,5}
%e A344653   96: {1,1,1,1,1,2}
%e A344653 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 A344653 Select[Range[100],Select[Permutations[Flatten[ConstantArray@@@FactorInteger[#]]],!MatchQ[#,{___,x_,y_,z_,___}/;x<=y<=z||x>=y>=z]&]=={}&]
%Y A344653 A superset of A335448, counted by A325535.
%Y A344653 Positions of 0's in A344606.
%Y A344653 These partitions are counted by A344654.
%Y A344653 The complement is A344742, counted by A344740.
%Y A344653 The separable case is A345173, counted by A345166.
%Y A344653 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A344653 A001250 counts wiggly permutations.
%Y A344653 A003242 counts anti-run compositions.
%Y A344653 A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
%Y A344653 A325534 counts separable partitions, ranked by A335433.
%Y A344653 A344604 counts wiggly compositions with twins.
%Y A344653 A345164 counts wiggly permutations of prime indices.
%Y A344653 A345165 counts partitions without a wiggly permutation, ranked by A345171.
%Y A344653 A345170 counts partitions with a wiggly permutation, ranked by A345172.
%Y A344653 A345192 counts non-wiggly compositions.
%Y A344653 Cf. A001222, A071321, A071322, A316523, A316524, A335126, A344614, A344615, A344616, A344652, A345163, A345168, A345193.
%K A344653 nonn
%O A344653 1,1
%A A344653 _Gus Wiseman_, Jun 12 2021