A351295 Numbers whose multiset of prime factors has no permutation with all distinct run-lengths.

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140

Offset: 1

Author

Gus Wiseman, Feb 16 2022

Keywords

• nonn

Comments

First differs from A130092 (non-Wilf partitions) in lacking 216.

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:
      6: (2,1)         46: (9,1)         84: (4,2,1,1)
     10: (3,1)         51: (7,2)         85: (7,3)
     14: (4,1)         55: (5,3)         86: (14,1)
     15: (3,2)         57: (8,2)         87: (10,2)
     21: (4,2)         58: (10,1)        90: (3,2,2,1)
     22: (5,1)         60: (3,2,1,1)     91: (6,4)
     26: (6,1)         62: (11,1)        93: (11,2)
     30: (3,2,1)       65: (6,3)         94: (15,1)
     33: (5,2)         66: (5,2,1)       95: (8,3)
     34: (7,1)         69: (9,2)        100: (3,3,1,1)
     35: (4,3)         70: (4,3,1)      102: (7,2,1)
     36: (2,2,1,1)     74: (12,1)       105: (4,3,2)
     38: (8,1)         77: (5,4)        106: (16,1)
     39: (6,2)         78: (6,2,1)      110: (5,3,1)
     42: (4,2,1)       82: (13,1)       111: (12,2)
For example, the prime indices of 150 are {1,2,3,3}, with permutations and run-lengths (right):
  (3,3,2,1) -> (2,1,1)
  (3,3,1,2) -> (2,1,1)
  (3,2,3,1) -> (1,1,1,1)
  (3,2,1,3) -> (1,1,1,1)
  (3,1,3,2) -> (1,1,1,1)
  (3,1,2,3) -> (1,1,1,1)
  (2,3,3,1) -> (1,2,1)
  (2,3,1,3) -> (1,1,1,1)
  (2,1,3,3) -> (1,1,2)
  (1,3,3,2) -> (1,2,1)
  (1,3,2,3) -> (1,1,1,1)
  (1,2,3,3) -> (1,1,2)
Since none have all distinct run-lengths, 150 is in the sequence.
		

Crossrefs

Wilf partitions are counted by A098859, ranked by A130091.

Non-Wilf partitions are counted by A336866, ranked by A130092.

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

These partitions appear to be counted by A351293.

The complement is A351294, apparently counted by A239455.

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 distinct run-lengths in binary expansion.

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

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

A297770 = number of distinct runs in binary expansion.

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

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

A329747 = runs-resistance, counted by A329746.

A333489 ranks anti-runs, complement A348612.

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

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

Programs

Mathematica

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

Extensions

Name edited by Gus Wiseman, Aug 13 2025