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A351295
Heinz numbers of non-Look-and-Say partitions. Numbers whose multiset of prime factors has no permutation with all distinct run-lengths.
5
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
1,1
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.
EXAMPLE
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.
MATHEMATICA
Select[Range[100], Select[Permutations[Join@@ ConstantArray@@@FactorInteger[#]], UnsameQ@@Length/@Split[#]&]=={}&]
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 are counted by A351293.
The complement is A351294, 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.
Sequence in context: A119847 A279458 A119899 * A362606 A130092 A289619
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 16 2022
STATUS
approved