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 A325853 #4 Jun 02 2019 00:49:11
%S A325853 1,1,2,3,5,7,11,14,21,28,39,51,69,88,116,148,193,242,309,385,484,596,
%T A325853 746,915,1128,1371,1679,2030,2460,2964,3570,4268,5115,6088,7251,8584,
%U A325853 10175,12002,14159,16619,19526,22846,26713,31153,36300,42169,48990,56728
%N A325853 Number of integer partitions of n such that every pair of distinct parts has a different quotient.
%C A325853 Also the number of integer partitions of n such that every orderless pair of (not necessarily distinct) parts has a different product.
%e A325853 The a(1) = 1 through a(7) = 14 partitions:
%e A325853 (1) (2) (3) (4) (5) (6) (7)
%e A325853 (11) (21) (22) (32) (33) (43)
%e A325853 (111) (31) (41) (42) (52)
%e A325853 (211) (221) (51) (61)
%e A325853 (1111) (311) (222) (322)
%e A325853 (2111) (321) (331)
%e A325853 (11111) (411) (511)
%e A325853 (2211) (2221)
%e A325853 (3111) (3211)
%e A325853 (21111) (4111)
%e A325853 (111111) (22111)
%e A325853 (31111)
%e A325853 (211111)
%e A325853 (1111111)
%e A325853 The one partition of 7 for which not every pair of distinct parts has a different quotient is (4,2,1).
%t A325853 Table[Length[Select[IntegerPartitions[n],UnsameQ@@Divide@@@Subsets[Union[#],{2}]&]],{n,0,20}]
%Y A325853 The subset case is A325860.
%Y A325853 The maximal case is A325861.
%Y A325853 The integer partition case is A325853.
%Y A325853 The strict integer partition case is A325854.
%Y A325853 Heinz numbers of the counterexamples are given by A325994.
%Y A325853 Cf. A002033, A103300, A108917, A143823, A196724, A325768, A325856, A325868, A325869, A325876.
%K A325853 nonn
%O A325853 0,3
%A A325853 _Gus Wiseman_, May 31 2019