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A342514
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Number of integer partitions of n with distinct first quotients.
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7
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1, 1, 2, 2, 4, 5, 6, 8, 11, 14, 18, 24, 28, 35, 41, 52, 64, 81, 93, 115, 137, 157, 190, 225, 268, 313, 366, 430, 502, 587, 683, 790, 913, 1055, 1217, 1393, 1605, 1830, 2098, 2384, 2722, 3101, 3524, 4005, 4524, 5137, 5812, 6570, 7434, 8360, 9416, 10602, 11881
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OFFSET
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0,3
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COMMENTS
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Also the number of reversed integer partitions of n with distinct first quotients.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).
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LINKS
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EXAMPLE
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The partition (4,3,3,2,1) has first quotients (3/4,1,2/3,1/2) so is counted under a(13), but it has first differences (-1,0,-1,-1) so is not counted under A325325(13).
The a(1) = 1 through a(9) = 14 partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(11) (21) (22) (32) (33) (43) (44) (54)
(31) (41) (42) (52) (53) (63)
(211) (221) (51) (61) (62) (72)
(311) (321) (322) (71) (81)
(411) (331) (332) (432)
(511) (422) (441)
(3211) (431) (522)
(521) (531)
(611) (621)
(3221) (711)
(3321)
(4311)
(5211)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@Divide@@@Partition[#, 2, 1]&]], {n, 0, 30}]
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CROSSREFS
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The version for differences instead of quotients is A325325.
The Heinz numbers of these partitions are A342521.
A000005 counts constant partitions.
A167865 counts strict chains of divisors > 1 summing to n.
A342096 counts partitions with all adjacent parts x < 2y (strict: A342097).
A342098 counts partitions with all adjacent parts x > 2y.
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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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