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Numbers whose prime factors have a permutation with no consecutive monotone triple, i.e., no triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.
21

%I #8 Jun 13 2021 10:21:35

%S 1,2,3,4,5,6,7,9,10,11,12,13,14,15,17,18,19,20,21,22,23,25,26,28,29,

%T 30,31,33,34,35,36,37,38,39,41,42,43,44,45,46,47,49,50,51,52,53,55,57,

%U 58,59,60,61,62,63,65,66,67,68,69,70,71,72,73,74,75,76,77

%N Numbers whose prime factors have a permutation with no consecutive monotone triple, i.e., no triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

%C Differs from A335433 in having all squares of primes (A001248) and lacking 270 etc.

%C Also Heinz numbers of integer partitions that are either a twin (x,x) or have a wiggly permutation.

%C (1) 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.

%C (2) 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).

%F Union of A345172 (wiggly) and A001248 (squares of primes).

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

%e 1: {} 18: {1,2,2} 36: {1,1,2,2}

%e 2: {1} 19: {8} 37: {12}

%e 3: {2} 20: {1,1,3} 38: {1,8}

%e 4: {1,1} 21: {2,4} 39: {2,6}

%e 5: {3} 22: {1,5} 41: {13}

%e 6: {1,2} 23: {9} 42: {1,2,4}

%e 7: {4} 25: {3,3} 43: {14}

%e 9: {2,2} 26: {1,6} 44: {1,1,5}

%e 10: {1,3} 28: {1,1,4} 45: {2,2,3}

%e 11: {5} 29: {10} 46: {1,9}

%e 12: {1,1,2} 30: {1,2,3} 47: {15}

%e 13: {6} 31: {11} 49: {4,4}

%e 14: {1,4} 33: {2,5} 50: {1,3,3}

%e 15: {2,3} 34: {1,7} 51: {2,7}

%e 17: {7} 35: {3,4} 52: {1,1,6}

%e For example, the prime factors of 120 are (2,2,2,3,5), with the two wiggly permutations (2,3,2,5,2) and (2,5,2,3,2), so 120 is in the sequence.

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

%Y Positions of nonzero terms in A344606.

%Y The complement is A344653, counted by A344654.

%Y These partitions are counted by A344740.

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

%Y A001248 lists squares of primes.

%Y A001250 counts wiggly permutations.

%Y A003242 counts anti-run compositions.

%Y A011782 counts compositions.

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

%Y A056239 adds up prime indices, row sums of A112798.

%Y A325534 counts separable partitions, ranked by A335433.

%Y A325535 counts inseparable partitions, ranked by A335448.

%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. A000070, A001222, A071321, A071322, A316523, A316524, A344605, A344614, A344616, A344652, A345163, A345166, A345167, A345173.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 12 2021