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 A342498 #11 Feb 16 2025 08:34:01
%S A342498 1,1,2,2,4,4,5,6,8,9,12,12,14,16,18,20,24,26,27,30,35,37,45,47,52,56,
%T A342498 61,65,72,77,83,90,95,99,109,117,127,135,144,151,164,172,181,197,209,
%U A342498 222,239,249,263,280,297,310,332,349,368,391,412,433,457,480,503
%N A342498 Number of integer partitions of n with strictly increasing first quotients.
%C A342498 Also the number of reversed integer partitions of n with strictly increasing first quotients.
%C A342498 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 A342498 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H A342498 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%H A342498 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%e A342498 The partition y = (13,7,2,1) has first quotients (7/13,2/7,1/2) so is not counted under a(23). However, the first differences (-6,-5,-1) are strictly increasing, so y is counted under A240027(23).
%e A342498 The a(1) = 1 through a(9) = 9 partitions:
%e A342498 (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e A342498 (11) (21) (22) (32) (33) (43) (44) (54)
%e A342498 (31) (41) (42) (52) (53) (63)
%e A342498 (211) (311) (51) (61) (62) (72)
%e A342498 (411) (322) (71) (81)
%e A342498 (511) (422) (522)
%e A342498 (521) (621)
%e A342498 (611) (711)
%e A342498 (5211)
%t A342498 Table[Length[Select[IntegerPartitions[n],Less@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]
%Y A342498 The version for differences instead of quotients is A240027.
%Y A342498 The ordered version is A342493.
%Y A342498 The weakly increasing version is A342497.
%Y A342498 The strictly decreasing version is A342499.
%Y A342498 The strict case is A342517.
%Y A342498 The Heinz numbers of these partitions are A342524.
%Y A342498 A000005 counts constant partitions.
%Y A342498 A000009 counts strict partitions.
%Y A342498 A000041 counts partitions.
%Y A342498 A001055 counts factorizations.
%Y A342498 A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A342498 A074206 counts ordered factorizations.
%Y A342498 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342498 A342096 counts partitions with adjacent x < 2y (strict: A342097).
%Y A342498 A342098 counts partitions with adjacent parts x > 2y.
%Y A342498 Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.
%K A342498 nonn
%O A342498 0,3
%A A342498 _Gus Wiseman_, Mar 17 2021