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
These partitions are counted by A351293.
A032020 = number of binary expansions with distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A165413 = number of distinct run-lengths in binary expansion.
A181819 = Heinz number of prime signature (prime shadow).
A297770 = number of distinct runs in binary expansion.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 16 2022
STATUS
approved