A345172 Numbers whose multiset of prime factors has an alternating permutation.

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 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, 78, 79, 82, 83

Offset

1,2

Comments

First differs from A212167 in containing 72.

First differs from A335433 in lacking 270, corresponding to the partition (3,2,2,2,1).

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,3,2,2,2,2,1) has no alternating permutations, even though it has the anti-run permutations (2,3,2,3,2,1,2), (2,3,2,1,2,3,2), and (2,1,2,3,2,3,2).

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.

Links

Table of n, a(n) for n=1..67.

Formula

Complement of A001248 (squares of primes) in A344742.

Example

The sequence of terms together with their prime indices begins:
      1: {}          20: {1,1,3}       39: {2,6}
      2: {1}         21: {2,4}         41: {13}
      3: {2}         22: {1,5}         42: {1,2,4}
      5: {3}         23: {9}           43: {14}
      6: {1,2}       26: {1,6}         44: {1,1,5}
      7: {4}         28: {1,1,4}       45: {2,2,3}
     10: {1,3}       29: {10}          46: {1,9}
     11: {5}         30: {1,2,3}       47: {15}
     12: {1,1,2}     31: {11}          50: {1,3,3}
     13: {6}         33: {2,5}         51: {2,7}
     14: {1,4}       34: {1,7}         52: {1,1,6}
     15: {2,3}       35: {3,4}         53: {16}
     17: {7}         36: {1,1,2,2}     55: {3,5}
     18: {1,2,2}     37: {12}          57: {2,8}
     19: {8}         38: {1,8}         58: {1,10}

Mathematica

wigQ[y_]:=Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1;
Select[Range[100], Select[Permutations[ Flatten[ConstantArray@@@FactorInteger[#]]], wigQ[#]&]!={}&]

Crossrefs

Including squares of primes A001248 gives A344742, counted by A344740.

This is a subset of A335433, which is counted by A325534.

Positions of nonzero terms in A345164.

The partitions with these Heinz numbers are counted by A345170.

The complement is A345171, which is counted by A345165.

A345173 = A345171 /\ A335433 is counted by A345166.

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

A001250 counts alternating permutations.

A003242 counts anti-run compositions.

A025047 counts alternating or wiggly compositions, also A025048, A025049.

A325535 counts inseparable partitions, ranked by A335448.

A344604 counts alternating compositions with twins.

A344606 counts alternating permutations of prime indices with twins.

A345192 counts non-alternating compositions.

Cf. A001222, A071321, A071322, A316523, A316524, A335126, A344605, A344614, A344616, A344653, A344654, A345163, A345167, A347706, A348379.

Sequence in context: A341646 A109421 A335433 * A212167 A339741 A317090

Adjacent sequences: A345169 A345170 A345171 * A345173 A345174 A345175

Keyword

nonn

Author

Gus Wiseman, Jun 13 2021

Status

approved