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A343381
Number of strict integer partitions of n with a part dividing all the others but no part divisible by all the others.
11
1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 6, 4, 9, 9, 14, 14, 20, 20, 30, 30, 39, 44, 59, 59, 77, 85, 106, 114, 145, 150, 191, 205, 247, 267, 328, 345, 418, 455, 544, 582, 699, 745, 886, 962, 1117, 1209, 1430, 1523, 1778, 1932, 2225, 2406, 2792, 3001, 3456, 3750
OFFSET
0,9
COMMENTS
Alternative name: Number of strict integer partitions of n that are empty or (1) have smallest part dividing all the others and (2) have greatest part not divisible by all the others.
EXAMPLE
The a(6) = 1 through a(16) = 14 partitions (empty column indicated by dot, A..D = 10..13):
321 . 431 531 541 641 642 751 761 861 862
521 721 731 651 5431 851 951 871
4321 5321 741 6421 941 A41 961
831 7321 A31 B31 A42
921 B21 6531 B41
5421 6431 7431 D21
6521 7521 6541
7421 9321 7531
8321 54321 7621
8431
8521
9421
A321
64321
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], #=={}||UnsameQ@@#&&And@@IntegerQ/@(#/Min@@#)&&!And@@IntegerQ/@(Max@@#/#)&]], {n, 0, 30}]
CROSSREFS
The first condition alone gives A097986.
The non-strict version is A343345 (Heinz numbers: A343340).
The second condition alone gives A343377.
The half-opposite versions are A343378 and A343379.
The opposite (and dual) version is A343380.
A000005 counts divisors.
A000009 counts strict 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: A124774 A056610 A341450 * A336096 A227774 A214920
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 16 2021
STATUS
approved