%I #9 Mar 22 2021 15:01:45
%S 1,1,2,3,7,13,19,36,67,114,197,322,564,976,1614,2729,4444,7364,12357,
%T 20231,33147
%N Number of compositions 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 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%e The composition (2,1,2,3) has first quotients (1/2,2,3/2) so is counted under a(8).
%e The a(1) = 1 through a(5) = 13 compositions:
%e (1) (2) (3) (4) (5)
%e (1,1) (1,2) (1,3) (1,4)
%e (2,1) (2,2) (2,3)
%e (3,1) (3,2)
%e (1,1,2) (4,1)
%e (1,2,1) (1,1,3)
%e (2,1,1) (1,2,2)
%e (1,3,1)
%e (2,1,2)
%e (2,2,1)
%e (3,1,1)
%e (1,1,2,1)
%e (1,2,1,1)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Divide@@@Partition[#,2,1]&]],{n,0,15}]
%Y The version for differences instead of quotients is A325545.
%Y The version for equal first quotients is A342495.
%Y The unordered version is A342514, ranked by A342521.
%Y The strict unordered version is A342520.
%Y A000005 counts constant compositions.
%Y A000009 counts strictly increasing (or strictly decreasing) compositions.
%Y A000041 counts weakly increasing (or weakly decreasing) compositions.
%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 Cf. A003242, A008965, A048004, A059966, A167606, A253249, A318991, A318992, A342527, A342528.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Mar 17 2021