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A342499
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Number of integer partitions of n with strictly decreasing first quotients.
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5
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1, 1, 2, 2, 3, 4, 5, 5, 7, 9, 10, 11, 14, 15, 18, 20, 23, 26, 31, 34, 39, 42, 45, 51, 58, 65, 70, 78, 83, 91, 102, 111, 122, 133, 145, 158, 170, 182, 202, 217, 231, 248, 268, 285, 307, 332, 354, 374, 404, 436, 468, 502, 537, 576, 618, 654, 694, 737, 782, 830
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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 partitions of n with strictly decreasing 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 (6,6,3,1) has first quotients (1,1/2,1/3) so is counted under a(16).
The a(1) = 1 through a(9) = 9 partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(11) (21) (22) (32) (33) (43) (44) (54)
(31) (41) (42) (52) (53) (63)
(221) (51) (61) (62) (72)
(321) (331) (71) (81)
(332) (432)
(431) (441)
(531)
(3321)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Greater@@Divide@@@Reverse/@Partition[#, 2, 1]&]], {n, 0, 30}]
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CROSSREFS
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The version for differences instead of quotients is A320470.
The strictly increasing version is A342498.
The weakly decreasing version is A342513.
The Heinz numbers of these partitions are listed by A342525.
A000005 counts constant partitions.
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A342098 counts partitions with 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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