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A342517
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Number of strict integer partitions of n with strictly increasing first quotients.
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5
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1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 8, 10, 11, 13, 14, 16, 16, 19, 21, 23, 27, 29, 31, 34, 36, 40, 43, 47, 49, 53, 56, 59, 66, 71, 75, 81, 86, 89, 97, 104, 110, 119, 123, 132, 143, 148, 156, 168, 177, 184, 198, 209, 218, 232, 246, 257, 269, 282, 294
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OFFSET
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0,4
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COMMENTS
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Also the number of reversed strict partitions of n with strictly increasing 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 (14,8,5,3,2) has first quotients (4/7,5/8,3/5,2/3) so is not counted under a(32), even though the differences (-6,-3,-2,-1) are strictly increasing.
The a(1) = 1 through a(13) = 10 partitions (A..D = 10..13):
1 2 3 4 5 6 7 8 9 A B C D
21 31 32 42 43 53 54 64 65 75 76
41 51 52 62 63 73 74 84 85
61 71 72 82 83 93 94
521 81 91 92 A2 A3
621 532 A1 B1 B2
721 632 732 C1
821 921 643
832
A21
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Less@@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 A179254.
The non-strict ordered version is A342493.
The weakly increasing version is A342516.
The strictly decreasing version is A342518.
A045690 counts sets with maximum n with all adjacent elements y < 2x.
A167865 counts strict chains of divisors > 1 summing to n.
A342096 counts partitions with all adjacent parts x < 2y (strict: A342097).
A342098 counts (strict) 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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