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A370804
Number of non-condensed integer partitions of n into parts > 1.
9
0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 3, 6, 6, 12, 14, 21, 25, 37, 43, 62, 75, 101, 124, 167, 198, 261, 316, 401, 488, 618
OFFSET
0,10
COMMENTS
These are partitions without ones such that it is not possible to choose a different divisor of each part.
EXAMPLE
The a(6) = 1 through a(14) = 12 partitions:
(222) . (2222) (333) (3322) (3332) (3333) (4333) (4442)
(3222) (4222) (5222) (4422) (7222) (5333)
(22222) (32222) (6222) (33322) (5522)
(33222) (43222) (8222)
(42222) (52222) (33332)
(222222) (322222) (43322)
(44222)
(53222)
(62222)
(332222)
(422222)
(2222222)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], FreeQ[#, 1] && Length[Select[Tuples[Divisors/@#], UnsameQ@@#&]]==0&]], {n, 0, 30}]
CROSSREFS
These partitions have as ranks the odd terms of A355740.
The version with ones is A370320, complement A239312.
The complement without ones is A370805.
The version for prime factors is A370807, with ones A370593.
The version for factorizations is A370813, complement A370814.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741, A355744, A355745 choose prime factors of prime indices.
Sequence in context: A301703 A143715 A159685 * A251729 A187763 A187262
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Mar 03 2024
STATUS
approved