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 A128699 Highly abundant numbers that are not superabundant, i.e., the complement of A004394 w.r.t. A002093. 2
 3, 8, 10, 16, 18, 20, 30, 42, 72, 84, 90, 96, 108, 144, 168, 210, 216, 288, 300, 336, 420, 480, 504, 540, 600, 630, 660, 960, 1008, 1080, 1200, 1440, 1560, 1620, 1800, 1920, 1980, 2100, 2160, 2340, 2400, 2880, 3024, 3120, 3240, 3360, 3600, 3780, 3960, 4200 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS In 1944, Alaoglu and Erdős conjectured that this sequence was infinite and this was proved to be true by Nicolas in 1969. LINKS L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Jean-Louis Nicolas, Ordre maximal d'un élément du groupe Sn des permutations et "highly composite numbers" Bull. Soc. Math. France 97: (1969), pp. 129-191. Eric Weisstein's World of Mathematics, Superabundant Number. Wikipedia, Highly Abundant Numbers. FORMULA 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). EXAMPLE 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. MATHEMATICA 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] CROSSREFS Cf. A000203, A002093, A004394, A128700, A128701, A128702. Sequence in context: A083246 A023492 A022801 * A104816 A181022 A185870 Adjacent sequences:  A128696 A128697 A128698 * A128700 A128701 A128702 KEYWORD nonn AUTHOR Ant King, Mar 28 2007 STATUS approved

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Last modified February 27 14:56 EST 2020. Contains 332306 sequences. (Running on oeis4.)