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Highly abundant numbers that are not superabundant, i.e., the complement of A004394 w.r.t. A002093.
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%I #19 Jul 22 2017 08:54:15

%S 3,8,10,16,18,20,30,42,72,84,90,96,108,144,168,210,216,288,300,336,

%T 420,480,504,540,600,630,660,960,1008,1080,1200,1440,1560,1620,1800,

%U 1920,1980,2100,2160,2340,2400,2880,3024,3120,3240,3360,3600,3780,3960,4200

%N Highly abundant numbers that are not superabundant, i.e., the complement of A004394 w.r.t. A002093.

%C In 1944, Alaoglu and Erdős conjectured that this sequence was infinite and this was proved to be true by Nicolas in 1969.

%H 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SuperabundantNumber.html">Superabundant Number</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Highly_abundant_number">Highly Abundant Numbers</a>.

%F 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 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 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 Cf. A000203, A002093, A004394, A128700, A128701, A128702.

%K nonn

%O 1,1

%A _Ant King_, Mar 28 2007