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A342520 Number of strict integer partitions of n with distinct first quotients.

%I A342520 #9 Feb 16 2025 08:34:01
%S A342520 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 A342520 127,147,170,198,225,258,297,340,385,448,499,566,647,737,832,937,1064,
%U A342520 1186,1348,1522,1701,1916,2157,2402,2697,3013,3355,3742,4190,4656,5191
%N A342520 Number of strict integer partitions of n with distinct first quotients.
%C A342520 Also the number of reversed strict integer partitions of n with distinct first quotients.
%C A342520 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 A342520 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H A342520 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%H A342520 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%e A342520 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 A342520 The a(1) = 1 through a(13) = 16 partitions (A..D = 10..13):
%e A342520   1   2   3    4    5    6     7    8     9     A      B      C     D
%e A342520           21   31   32   42    43   53    54    64     65     75    76
%e A342520                     41   51    52   62    63    73     74     84    85
%e A342520                          321   61   71    72    82     83     93    94
%e A342520                                     431   81    91     92     A2    A3
%e A342520                                     521   432   532    A1     B1    B2
%e A342520                                           531   541    542    543   C1
%e A342520                                           621   631    632    642   643
%e A342520                                                 721    641    651   652
%e A342520                                                 4321   731    732   742
%e A342520                                                        821    741   751
%e A342520                                                        5321   831   832
%e A342520                                                               921   841
%e A342520                                                                     A21
%e A342520                                                                     5431
%e A342520                                                                     7321
%t A342520 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Divide@@@Partition[#,2,1]&]],{n,0,30}]
%Y A342520 The version for differences instead of quotients is A320347.
%Y A342520 The non-strict version is A342514 (ranking: A342521).
%Y A342520 The equal instead of distinct version is A342515.
%Y A342520 The non-strict ordered version is A342529.
%Y A342520 The version for strict divisor chains is A342530.
%Y A342520 A000041 counts partitions (strict: A000009).
%Y A342520 A001055 counts factorizations (strict: A045778, ordered: A074206).
%Y A342520 A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A342520 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342520 A342086 counts strict chains of divisors with strictly increasing quotients.
%Y A342520 A342098 counts (strict) partitions with all adjacent parts x > 2y.
%Y A342520 Cf. A000005, A003114, A003242, A005117, A018819, A067824, A238710, A253249, A318991, A318992.
%K A342520 nonn
%O A342520 0,4
%A A342520 _Gus Wiseman_, Mar 20 2021