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%I #11 Feb 21 2023 12:28:48
%S 1,1,2,2,4,5,6,8,11,14,18,24,28,35,41,52,64,81,93,115,137,157,190,225,
%T 268,313,366,430,502,587,683,790,913,1055,1217,1393,1605,1830,2098,
%U 2384,2722,3101,3524,4005,4524,5137,5812,6570,7434,8360,9416,10602,11881
%N Number of integer partitions of n with distinct first quotients.
%C Also the number of reversed 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 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).
%e The a(1) = 1 through a(9) = 14 partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e (11) (21) (22) (32) (33) (43) (44) (54)
%e (31) (41) (42) (52) (53) (63)
%e (211) (221) (51) (61) (62) (72)
%e (311) (321) (322) (71) (81)
%e (411) (331) (332) (432)
%e (511) (422) (441)
%e (3211) (431) (522)
%e (521) (531)
%e (611) (621)
%e (3221) (711)
%e (3321)
%e (4311)
%e (5211)
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@Divide@@@Partition[#,2,1]&]],{n,0,30}]
%Y The version for differences instead of quotients is A325325.
%Y The ordered version is A342529.
%Y The strict case is A342520.
%Y The Heinz numbers of these partitions are A342521.
%Y A000005 counts constant partitions.
%Y A000009 counts strict partitions.
%Y A000041 counts partitions.
%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 A342096 counts partitions with all adjacent parts x < 2y (strict: A342097).
%Y A342098 counts partitions with all adjacent parts x > 2y.
%Y Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.
%K nonn
%O 0,3
%A _Gus Wiseman_, Mar 17 2021
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