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A342529 Number of compositions of n with distinct first quotients.

%I A342529 #14 Jan 16 2025 21:01:16
%S A342529 1,1,2,3,7,13,19,36,67,114,197,322,564,976,1614,2729,4444,7364,12357,
%T A342529 20231,33147,53973,87254,140861,227535,368050,589706,940999,1497912,
%U A342529 2378260,3774297,5964712,9416411,14822087,23244440,36420756
%N A342529 Number of compositions of n with distinct first quotients.
%C A342529 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 A342529 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%e A342529 The composition (2,1,2,3) has first quotients (1/2,2,3/2) so is counted under a(8).
%e A342529 The a(1) = 1 through a(5) = 13 compositions:
%e A342529   (1)  (2)    (3)    (4)      (5)
%e A342529        (1,1)  (1,2)  (1,3)    (1,4)
%e A342529               (2,1)  (2,2)    (2,3)
%e A342529                      (3,1)    (3,2)
%e A342529                      (1,1,2)  (4,1)
%e A342529                      (1,2,1)  (1,1,3)
%e A342529                      (2,1,1)  (1,2,2)
%e A342529                               (1,3,1)
%e A342529                               (2,1,2)
%e A342529                               (2,2,1)
%e A342529                               (3,1,1)
%e A342529                               (1,1,2,1)
%e A342529                               (1,2,1,1)
%t A342529 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Divide@@@Partition[#,2,1]&]],{n,0,15}]
%Y A342529 The version for differences instead of quotients is A325545.
%Y A342529 The version for equal first quotients is A342495.
%Y A342529 The unordered version is A342514, ranked by A342521.
%Y A342529 The strict unordered version is A342520.
%Y A342529 A000005 counts constant compositions.
%Y A342529 A000009 counts strictly increasing (or strictly decreasing) compositions.
%Y A342529 A000041 counts weakly increasing (or weakly decreasing) compositions.
%Y A342529 A001055 counts factorizations (strict: A045778, ordered: A074206).
%Y A342529 A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A342529 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342529 Cf. A003242, A008965, A048004, A059966, A167606, A253249, A318991, A318992, A342527, A342528.
%K A342529 nonn,more
%O A342529 0,3
%A A342529 _Gus Wiseman_, Mar 17 2021
%E A342529 a(21)-a(35) from _Alois P. Heinz_, Jan 16 2025