cp's OEIS Frontend

A344742 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.

%I A344742 #8 Jun 13 2021 10:21:35
%S A344742 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 A344742 30,31,33,34,35,36,37,38,39,41,42,43,44,45,46,47,49,50,51,52,53,55,57,
%U A344742 58,59,60,61,62,63,65,66,67,68,69,70,71,72,73,74,75,76,77
%N A344742 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 A344742 Differs from A335433 in having all squares of primes (A001248) and lacking 270 etc.
%C A344742 Also Heinz numbers of integer partitions that are either a twin (x,x) or have a wiggly permutation.
%C A344742 (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 A344742 (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 A344742 Union of A345172 (wiggly) and A001248 (squares of primes).
%e A344742 The sequence of terms together with their prime indices begins:
%e A344742       1: {}          18: {1,2,2}     36: {1,1,2,2}
%e A344742       2: {1}         19: {8}         37: {12}
%e A344742       3: {2}         20: {1,1,3}     38: {1,8}
%e A344742       4: {1,1}       21: {2,4}       39: {2,6}
%e A344742       5: {3}         22: {1,5}       41: {13}
%e A344742       6: {1,2}       23: {9}         42: {1,2,4}
%e A344742       7: {4}         25: {3,3}       43: {14}
%e A344742       9: {2,2}       26: {1,6}       44: {1,1,5}
%e A344742      10: {1,3}       28: {1,1,4}     45: {2,2,3}
%e A344742      11: {5}         29: {10}        46: {1,9}
%e A344742      12: {1,1,2}     30: {1,2,3}     47: {15}
%e A344742      13: {6}         31: {11}        49: {4,4}
%e A344742      14: {1,4}       33: {2,5}       50: {1,3,3}
%e A344742      15: {2,3}       34: {1,7}       51: {2,7}
%e A344742      17: {7}         35: {3,4}       52: {1,1,6}
%e A344742 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 A344742 Select[Range[100],Select[Permutations[Flatten[ConstantArray@@@FactorInteger[#]]],!MatchQ[#,{___,x_,y_,z_,___}/;x<=y<=z||x>=y>=z]&]!={}&]
%Y A344742 Positions of nonzero terms in A344606.
%Y A344742 The complement is A344653, counted by A344654.
%Y A344742 These partitions are counted by A344740.
%Y A344742 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A344742 A001248 lists squares of primes.
%Y A344742 A001250 counts wiggly permutations.
%Y A344742 A003242 counts anti-run compositions.
%Y A344742 A011782 counts compositions.
%Y A344742 A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
%Y A344742 A056239 adds up prime indices,  row sums of A112798.
%Y A344742 A325534 counts separable partitions, ranked by A335433.
%Y A344742 A325535 counts inseparable partitions, ranked by A335448.
%Y A344742 A344604 counts wiggly compositions with twins.
%Y A344742 A345164 counts wiggly permutations of prime indices.
%Y A344742 A345165 counts partitions without a wiggly permutation, ranked by A345171.
%Y A344742 A345170 counts partitions with a wiggly permutation, ranked by A345172.
%Y A344742 A345192 counts non-wiggly compositions.
%Y A344742 Cf. A000070, A001222, A071321, A071322, A316523, A316524, A344605, A344614, A344616, A344652, A345163, A345166, A345167, A345173.
%K A344742 nonn
%O A344742 1,2
%A A344742 _Gus Wiseman_, Jun 12 2021