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A371178
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Number of integer partitions of n containing all divisors of all parts.
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11
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1, 1, 1, 2, 3, 4, 6, 9, 12, 16, 21, 28, 37, 48, 62, 80, 101, 127, 162, 202, 252, 312, 386, 475, 585, 713, 869, 1056, 1278, 1541, 1859, 2232, 2675, 3196, 3811, 4534, 5386, 6379, 7547, 8908, 10497, 12345, 14501, 16999, 19897, 23253, 27135, 31618, 36796, 42756
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OFFSET
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0,4
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COMMENTS
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The Heinz numbers of these partitions are given by A371177.
Also partitions such that the number of distinct parts is equal to the number of distinct divisors of parts.
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LINKS
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EXAMPLE
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The partition (4,2,1,1) contains all distinct divisors {1,2,4}, so is counted under a(8).
The partition (4,4,3,2,2,2,1) contains all distinct divisors {1,2,3,4} so is counted under 4 + 4 + 3 + 2 + 2 + 2 + 1 = 18. - David A. Corneth, Mar 18 2024
The a(0) = 1 through a(8) = 12 partitions:
() (1) (11) (21) (31) (221) (51) (331) (71)
(111) (211) (311) (321) (421) (521)
(1111) (2111) (2211) (511) (3221)
(11111) (3111) (2221) (3311)
(21111) (3211) (4211)
(111111) (22111) (5111)
(31111) (22211)
(211111) (32111)
(1111111) (221111)
(311111)
(2111111)
(11111111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], SubsetQ[#, Union@@Divisors/@#]&]], {n, 0, 30}]
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CROSSREFS
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For partitions with no divisors of parts we have A305148, ranks A316476.
The complement is counted by A371132.
For submultisets instead of distinct parts we have A371172, ranks A371165.
These partitions have ranks A371177.
A008284 counts partitions by length.
Cf. A000837, A003963, A239312, A285573, A305148, A319055, A355529, A370803, A370808, A370813, A371168, A371171, A371173.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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