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A342530 Number of strict chains of divisors ending with n and having distinct first quotients.

%I A342530 #12 Jun 17 2025 09:18:38
%S A342530 1,2,2,3,2,6,2,6,3,6,2,12,2,6,6,9,2,12,2,12,6,6,2,28,3,6,6,12,2,26,2,
%T A342530 14,6,6,6,31,2,6,6,28,2,26,2,12,12,6,2,52,3,12,6,12,2,28,6,28,6,6,2,
%U A342530 66,2,6,12,25,6,26,2,12,6,26,2,76,2,6,12,12,6,26
%N A342530 Number of strict chains of divisors ending with n and having distinct first quotients.
%C A342530 The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the quotients of (6,3,1) are (1/2,1/3).
%H A342530 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%F A342530 a(n) = Sum_{d|n} A254578(d). - _Ridouane Oudra_, Jun 17 2025
%e A342530 The a(1) = 1 through a(12) = 12 chains (reversed):
%e A342530   1  2    3    4    5    6      7    8      9    10      11    12
%e A342530      2/1  3/1  4/1  5/1  6/1    7/1  8/1    9/1  10/1    11/1  12/1
%e A342530                4/2       6/2         8/2    9/3  10/2          12/2
%e A342530                          6/3         8/4         10/5          12/3
%e A342530                          6/2/1       8/2/1       10/2/1        12/4
%e A342530                          6/3/1       8/4/1       10/5/1        12/6
%e A342530                                                                12/2/1
%e A342530                                                                12/3/1
%e A342530                                                                12/4/1
%e A342530                                                                12/4/2
%e A342530                                                                12/6/1
%e A342530                                                                12/6/2
%e A342530 Not counted under a(12) are: 12/4/2/1, 12/6/2/1, 12/6/3, 12/6/3/1.
%t A342530 cmi[n_]:=Prepend[Prepend[#,n]&/@Join@@cmi/@Most[Divisors[n]],{n}];
%t A342530 Table[Length[Select[cmi[n],UnsameQ@@Divide@@@Partition[#,2,1]&]],{n,100}]
%Y A342530 The version for weakly increasing first quotients is A057567.
%Y A342530 The version for equal first quotients is A169594.
%Y A342530 The case of chains starting with 1 is A254578.
%Y A342530 The version for strictly increasing first quotients is A342086.
%Y A342530 A001055 counts factorizations (strict: A045778, ordered: A074206).
%Y A342530 A067824 counts strict chains of divisors ending with n.
%Y A342530 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342530 A253249 counts strict chains of divisors.
%Y A342530 A334997 counts chains of divisors of n by length.
%Y A342530 A342495/A342529 count compositions with equal/distinct quotients.
%Y A342530 A342496/A342514 count partitions with equal/distinct quotients.
%Y A342530 A342515/A342520 count strict partitions with equal/distinct quotients.
%Y A342530 A342522/A342521 rank partitions with equal/distinct quotients.
%Y A342530 Cf. A000009, A003238, A003242, A122651, A179254, A318991, A318992, A325545.
%K A342530 nonn
%O A342530 1,2
%A A342530 _Gus Wiseman_, Mar 25 2021