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 A128699 #26 Feb 16 2025 08:33:05
%S A128699 3,8,10,16,18,20,30,42,72,84,90,96,108,144,168,210,216,288,300,336,
%T A128699 420,480,504,540,600,630,660,960,1008,1080,1200,1440,1560,1620,1800,
%U A128699 1920,1980,2100,2160,2340,2400,2880,3024,3120,3240,3360,3600,3780,3960,4200
%N A128699 Highly abundant numbers that are not superabundant, i.e., the complement of A004394 w.r.t. A002093.
%C A128699 In 1944, Alaoglu and Erdős conjectured that this sequence was infinite and this was proved to be true by Nicolas in 1969.
%H A128699 Amiram Eldar, <a href="/A128699/b128699.txt">Table of n, a(n) for n = 1..1000</a>
%H A128699 L. Alaoglu and P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1944-03.pdf">On highly composite and similar numbers</a>, Trans. Amer. Math. Soc., 56 (1944), 448-469.
%H A128699 Jean-Louis Nicolas, <a href="http://archive.numdam.org/article/BSMF_1969__97__129_0.pdf">Ordre maximal d'un élément du groupe Sn des permutations et "highly composite numbers"</a> Bull. Soc. Math. France 97 (1969), pp. 129-191.
%H A128699 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SuperabundantNumber.html">Superabundant Number</a>.
%H A128699 Wikipedia, <a href="http://en.wikipedia.org/wiki/Highly_abundant_number">Highly Abundant Numbers</a>.
%F A128699 The highly abundant numbers are those integers for which sigma(n) > sigma(m) for all m < n (A002093) and the superabundant numbers are those integers for which sigma(n)/n > sigma(m)/m for all m < n (A004394).
%e A128699 The sequence of highly abundant numbers begins 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20 and the sequence of superabundant numbers begins 1, 2, 4, 6, 12, 24. Because 10 is the third number which is in the first sequence but not in the second, it follows that a(3)=10.
%t A128699 habdata1=FoldList[Max,1,Table[DivisorSigma[1,n],{n,2,10000}]]; data1=Flatten[Position[habdata1,#,1,1]&/@Union[habdata1]];sabdata2=FoldList[Max,1,Table[DivisorSigma[1,n]/n,{n,2,10000}]]; data2=Flatten[Position[sabdata2,#,1,1]&/@Union[sabdata2]];sabdata2=FoldList[Max,1,Table[DivisorSigma[1,n]/n,{n,2,10000}]]; Complement[data1,data2]
%Y A128699 Cf. A000203, A002093, A004394, A128700, A128701, A128702.
%K A128699 nonn
%O A128699 1,1
%A A128699 _Ant King_, Mar 28 2007