A351294 Numbers whose multiset of prime factors has at least one permutation with all distinct run-lengths.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109

Offset

1,2

Comments

First differs from A130091 (Wilf partitions) in having 216.

See A239455 for the definition of Look-and-Say partitions.

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..66.

Example

The terms together with their prime indices begin:
      1: ()            20: (3,1,1)         47: (15)
      2: (1)           23: (9)             48: (2,1,1,1,1)
      3: (2)           24: (2,1,1,1)       49: (4,4)
      4: (1,1)         25: (3,3)           50: (3,3,1)
      5: (3)           27: (2,2,2)         52: (6,1,1)
      7: (4)           28: (4,1,1)         53: (16)
      8: (1,1,1)       29: (10)            54: (2,2,2,1)
      9: (2,2)         31: (11)            56: (4,1,1,1)
     11: (5)           32: (1,1,1,1,1)     59: (17)
     12: (2,1,1)       37: (12)            61: (18)
     13: (6)           40: (3,1,1,1)       63: (4,2,2)
     16: (1,1,1,1)     41: (13)            64: (1,1,1,1,1,1)
     17: (7)           43: (14)            67: (19)
     18: (2,2,1)       44: (5,1,1)         68: (7,1,1)
     19: (8)           45: (3,2,2)         71: (20)
For example, the prime indices of 216 are {1,1,1,2,2,2}, and there are four permutations with distinct run-lengths: (1,1,2,2,2,1), (1,2,2,2,1,1), (2,1,1,1,2,2), (2,2,1,1,1,2); so 216 is in the sequence. It is the Heinz number of the Look-and-Say partition of (3,3,2,1).

Mathematica

Select[Range[100], Select[Permutations[Join@@ ConstantArray@@@FactorInteger[#]], UnsameQ@@Length/@Split[#]&]!={}&]

Crossrefs

The Wilf case (distinct multiplicities) is A130091, counted by A098859.

The complement of the Wilf case is A130092, counted by A336866.

These partitions appear to be counted by A239455.

A variant for runs is A351201, counted by A351203 (complement A351204).

The complement is A351295, counted by A351293.

A032020 = number of binary expansions with distinct run-lengths.

A044813 = numbers whose binary expansion has all distinct run-lengths.

A056239 = sum of prime indices, row sums of A112798.

A165413 = number of run-lengths in binary expansion, for all runs A297770.

A181819 = Heinz number of prime signature (prime shadow).

A182850/A323014 = frequency depth, counted by A225485/A325280.

A320922 ranks graphical partitions, complement A339618, counted by A000569.

A329739 = compositions with all distinct run-lengths, for all runs A351013.

A333489 ranks anti-runs, complement A348612.

A351017 = binary words with all distinct run-lengths, for all runs A351016.

A351292 = patterns with all distinct run-lengths, for all runs A351200.

Cf. A000961, A001221, A001222, A047966, A175413, A182857, A304660, A320924, A328592, A329747, A351202, A351290, A351592.

Sequence in context: A325370 A329139 A356862 * A383091 A130091 A359178

Adjacent sequences: A351291 A351292 A351293 * A351295 A351296 A351297

Keyword

nonn

Author

Gus Wiseman, Feb 15 2022

Extensions

Name edited by Gus Wiseman, Aug 13 2025

Status

approved