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Number of strict integer partitions of n with distinct first quotients.
6

%I #8 Mar 22 2021 15:01:39

%S 1,1,1,2,2,3,4,4,6,8,10,12,13,16,20,25,30,37,42,50,57,65,80,93,108,

%T 127,147,170,198,225,258,297,340,385,448,499,566,647,737,832,937,1064,

%U 1186,1348,1522,1701,1916,2157,2402,2697,3013,3355,3742,4190,4656,5191

%N Number of strict integer partitions of n with distinct first quotients.

%C Also the number of reversed strict integer partitions of n with distinct first quotients.

%C 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).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.

%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.

%H Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.

%e The strict partition (12,10,5,2,1) has first quotients (5/6,1/2,2/5,1/2) so is not counted under a(30), even though the first differences (-2,-5,-3,-1) are distinct.

%e The a(1) = 1 through a(13) = 16 partitions (A..D = 10..13):

%e 1 2 3 4 5 6 7 8 9 A B C D

%e 21 31 32 42 43 53 54 64 65 75 76

%e 41 51 52 62 63 73 74 84 85

%e 321 61 71 72 82 83 93 94

%e 431 81 91 92 A2 A3

%e 521 432 532 A1 B1 B2

%e 531 541 542 543 C1

%e 621 631 632 642 643

%e 721 641 651 652

%e 4321 731 732 742

%e 821 741 751

%e 5321 831 832

%e 921 841

%e A21

%e 5431

%e 7321

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Divide@@@Partition[#,2,1]&]],{n,0,30}]

%Y The version for differences instead of quotients is A320347.

%Y The non-strict version is A342514 (ranking: A342521).

%Y The equal instead of distinct version is A342515.

%Y The non-strict ordered version is A342529.

%Y The version for strict divisor chains is A342530.

%Y A000041 counts partitions (strict: A000009).

%Y A001055 counts factorizations (strict: A045778, ordered: A074206).

%Y A003238 counts chains of divisors summing to n - 1 (strict: A122651).

%Y A167865 counts strict chains of divisors > 1 summing to n.

%Y A342086 counts strict chains of divisors with strictly increasing quotients.

%Y A342098 counts (strict) partitions with all adjacent parts x > 2y.

%Y Cf. A000005, A003114, A003242, A005117, A018819, A067824, A238710, A253249, A318991, A318992.

%K nonn

%O 0,4

%A _Gus Wiseman_, Mar 20 2021