A351201 Numbers whose multiset of prime factors has a permutation without all distinct runs.

12, 18, 20, 28, 36, 44, 45, 48, 50, 52, 60, 63, 68, 72, 75, 76, 80, 84, 90, 92, 98, 99, 100, 108, 112, 116, 117, 120, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 162, 164, 168, 171, 172, 175, 176, 180, 188, 192, 196, 198, 200, 204, 207, 208, 212, 216

Offset: 1

Author

Gus Wiseman, Feb 12 2022

Keywords

• nonn

Examples

			The prime factors of 80 are {2,2,2,2,5} and the permutation (2,2,5,2,2) has runs (2,2), (5), and (2,2), which are not all distinct, so 80 is in the sequence. On the other hand, 24 has prime factors {2,2,2,3}, and all four permutations (3,2,2,2), (2,3,2,2), (2,2,3,2), (2,2,2,3) have distinct runs, so 24 is not in the sequence.
The terms and their prime indices begin:
     12: (2,1,1)         76: (8,1,1)        132: (5,2,1,1)
     18: (2,2,1)         80: (3,1,1,1,1)    140: (4,3,1,1)
     20: (3,1,1)         84: (4,2,1,1)      144: (2,2,1,1,1,1)
     28: (4,1,1)         90: (3,2,2,1)      147: (4,4,2)
     36: (2,2,1,1)       92: (9,1,1)        148: (12,1,1)
     44: (5,1,1)         98: (4,4,1)        150: (3,3,2,1)
     45: (3,2,2)         99: (5,2,2)        153: (7,2,2)
     48: (2,1,1,1,1)    100: (3,3,1,1)      156: (6,2,1,1)
     50: (3,3,1)        108: (2,2,2,1,1)    162: (2,2,2,2,1)
     52: (6,1,1)        112: (4,1,1,1,1)    164: (13,1,1)
     60: (3,2,1,1)      116: (10,1,1)       168: (4,2,1,1,1)
     63: (4,2,2)        117: (6,2,2)        171: (8,2,2)
     68: (7,1,1)        120: (3,2,1,1,1)    172: (14,1,1)
     72: (2,2,1,1,1)    124: (11,1,1)       175: (4,3,3)
     75: (3,3,2)        126: (4,2,2,1)      176: (5,1,1,1,1)
		

Links

• Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Crossrefs

The version for run-lengths instead of runs is A024619.

These permutations are counted by A351202.

These rank the partitions counted by A351203, complement A351204.

A005811 counts runs in binary expansion.

A044813 lists numbers whose binary expansion has distinct run-lengths.

A056239 adds up prime indices, row sums of A112798.

A283353 counts normal multisets with a permutation w/o all distinct runs.

A297770 counts distinct runs in binary expansion.

A333489 ranks anti-runs, complement A348612.

A351014 counts distinct runs in standard compositions, firsts A351015.

A351291 ranks compositions without all distinct runs.

Counting words with all distinct runs:

- A351013 = compositions, for run-lengths A329739, ranked by A351290.

- A351016 = binary words, for run-lengths A351017.

- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.

- A351200 = patterns, for run-lengths A351292.

Cf. A001055, A001221, A001222, A061395, A098859, A106356, A106529.

Programs

Mathematica

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