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A351294
Heinz numbers of Look-and-Say partitions. Numbers whose multiset of prime factors has at least one permutation with all distinct run-lengths.
11
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.
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 are 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.
Sequence in context: A325370 A329139 A356862 * A130091 A359178 A344609
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 15 2022
STATUS
approved