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

Author

Gus Wiseman, Feb 15 2022

Keywords

• nonn

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.

Examples

			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).
		

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.

Programs

Mathematica

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

Extensions

Name edited by Gus Wiseman, Aug 13 2025