|
%I #8 Mar 25 2021 12:44:13
%S 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 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 66,2,6,12,25,6,26,2,12,6,26,2,76,2,6,12,12,6,26
%N Number of strict chains of divisors ending with n and having 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 quotients of (6,3,1) are (1/2,1/3).
%H Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%e The a(1) = 1 through a(12) = 12 chains (reversed):
%e 1 2 3 4 5 6 7 8 9 10 11 12
%e 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 10/1 11/1 12/1
%e 4/2 6/2 8/2 9/3 10/2 12/2
%e 6/3 8/4 10/5 12/3
%e 6/2/1 8/2/1 10/2/1 12/4
%e 6/3/1 8/4/1 10/5/1 12/6
%e 12/2/1
%e 12/3/1
%e 12/4/1
%e 12/4/2
%e 12/6/1
%e 12/6/2
%e Not counted under a(12) are: 12/4/2/1, 12/6/2/1, 12/6/3, 12/6/3/1.
%t cmi[n_]:=Prepend[Prepend[#,n]&/@Join@@cmi/@Most[Divisors[n]],{n}];
%t Table[Length[Select[cmi[n],UnsameQ@@Divide@@@Partition[#,2,1]&]],{n,100}]
%Y The version for weakly increasing first quotients is A057567.
%Y The version for equal first quotients is A169594.
%Y The case of chains starting with 1 is A254578.
%Y The version for strictly increasing first quotients is A342086.
%Y A001055 counts factorizations (strict: A045778, ordered: A074206).
%Y A067824 counts strict chains of divisors ending with n.
%Y A167865 counts strict chains of divisors > 1 summing to n.
%Y A253249 counts strict chains of divisors.
%Y A334997 counts chains of divisors of n by length.
%Y A342495/A342529 count compositions with equal/distinct quotients.
%Y A342496/A342514 count partitions with equal/distinct quotients.
%Y A342515/A342520 count strict partitions with equal/distinct quotients.
%Y A342522/A342521 rank partitions with equal/distinct quotients.
%Y Cf. A000009, A003238, A003242, A122651, A179254, A318991, A318992, A325545.
%K nonn
%O 1,2
%A _Gus Wiseman_, Mar 25 2021
|
