This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A342495 #16 Feb 16 2025 08:34:01
%S A342495 1,1,2,4,5,6,8,10,10,11,12,12,16,16,18,20,19,18,22,22,24,28,24,24,30,
%T A342495 27,30,30,34,30,38,36,36,36,36,40,43,40,42,46,48,42,52,46,48,52,48,48,
%U A342495 56,55,54,54,58,54,60,58,64,64,60,60,72,64,68,74,69,72,72
%N A342495 Number of compositions of n with constant (equal) first quotients.
%C A342495 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 A342495 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H A342495 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%F A342495 a(n > 0) = 2*A342496(n) - A000005(n).
%e A342495 The composition (1,2,4,8) has first quotients (2,2,2) so is counted under a(15).
%e A342495 The composition (4,5,6) has first quotients (5/4,6/5) so is not counted under a(15).
%e A342495 The a(1) = 1 through a(7) = 10 compositions:
%e A342495 (1) (2) (3) (4) (5) (6) (7)
%e A342495 (11) (12) (13) (14) (15) (16)
%e A342495 (21) (22) (23) (24) (25)
%e A342495 (111) (31) (32) (33) (34)
%e A342495 (1111) (41) (42) (43)
%e A342495 (11111) (51) (52)
%e A342495 (222) (61)
%e A342495 (111111) (124)
%e A342495 (421)
%e A342495 (1111111)
%t A342495 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Divide@@@Partition[#,2,1]&]],{n,0,15}]
%Y A342495 The version for differences instead of quotients is A175342.
%Y A342495 The unordered version is A342496, ranked by A342522.
%Y A342495 The strict unordered version is A342515.
%Y A342495 The distinct version is A342529.
%Y A342495 A000005 counts constant compositions.
%Y A342495 A000009 counts strictly increasing (or strictly decreasing) compositions.
%Y A342495 A000041 counts weakly increasing (or weakly decreasing) compositions.
%Y A342495 A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A342495 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342495 Cf. A002843, A003242, A008965, A048004, A059966, A074206, A167606, A253249, A318991, A318992, A325557, A342528.
%K A342495 nonn
%O A342495 0,3
%A A342495 _Gus Wiseman_, Mar 17 2021