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.

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, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Author

Gus Wiseman, Jun 12 2021

Keywords

• nonn

Comments

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

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

(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.

(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).

Examples

			The sequence of terms together with their prime indices begins:
      1: {}          18: {1,2,2}     36: {1,1,2,2}
      2: {1}         19: {8}         37: {12}
      3: {2}         20: {1,1,3}     38: {1,8}
      4: {1,1}       21: {2,4}       39: {2,6}
      5: {3}         22: {1,5}       41: {13}
      6: {1,2}       23: {9}         42: {1,2,4}
      7: {4}         25: {3,3}       43: {14}
      9: {2,2}       26: {1,6}       44: {1,1,5}
     10: {1,3}       28: {1,1,4}     45: {2,2,3}
     11: {5}         29: {10}        46: {1,9}
     12: {1,1,2}     30: {1,2,3}     47: {15}
     13: {6}         31: {11}        49: {4,4}
     14: {1,4}       33: {2,5}       50: {1,3,3}
     15: {2,3}       34: {1,7}       51: {2,7}
     17: {7}         35: {3,4}       52: {1,1,6}
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.
		

Crossrefs

Positions of nonzero terms in A344606.

The complement is A344653, counted by A344654.

These partitions are counted by A344740.

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

A001248 lists squares of primes.

A001250 counts wiggly permutations.

A003242 counts anti-run compositions.

A011782 counts compositions.

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

A056239 adds up prime indices, row sums of A112798.

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

A344604 counts wiggly compositions with twins.

A345164 counts wiggly permutations of prime indices.

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

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

A345192 counts non-wiggly compositions.

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

Programs

Mathematica

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

Formula

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