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A343380
Number of strict integer partitions of n with no part dividing all the others but with a part divisible by all the others.
10
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 1, 1, 4, 0, 1, 0, 2, 0, 4, 0, 3, 1, 2, 2, 5, 0, 5, 3, 4, 1, 9, 1, 5, 2, 4, 5, 11, 1, 6, 4, 11, 3, 13, 5, 10, 4, 11, 8, 14, 3, 10, 6, 9, 3, 15, 6, 14, 10, 18, 8
OFFSET
0,18
COMMENTS
Alternative name: Number of strict integer partitions of n that are either empty or (1) have smallest part not dividing all the others and (2) have greatest part divisible by all the others.
EXAMPLE
The a(11) = 1 through a(29) = 4 partitions (empty columns indicated by dots, A..O = 10..24):
632 . . . . . A52 . C43 . C432 C64 E72 . C643 . K52 . I92
C32 F53 C6432 K54
I32 O32
C632 I632
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], #=={}||UnsameQ@@#&&!And@@IntegerQ/@(#/Min@@#)&&And@@IntegerQ/@(Max@@#/#)&]], {n, 0, 30}]
CROSSREFS
The first condition alone gives A341450.
The non-strict version is A343344 (Heinz numbers: A343339).
The second condition alone gives A343347.
The half-opposite versions are A343378 and A343379.
The opposite (and dual) version is A343381.
A000009 counts strict partitions.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Sequence in context: A091830 A029427 A353506 * A132343 A277731 A298307
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 16 2021
STATUS
approved