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Search: id:A128699
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| A128699 |
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Highly abundant numbers that are not superabundant i.e. The complement of A002093 and A004394. |
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3
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| 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; listen)
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OFFSET
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1,1
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COMMENT
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In 1944, Alaoglu and Erdos conjectured that this sequence was infinite, and this was proved to be true by Nicolas in 1969.
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REFERENCES
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Alaoglu, L., Erdos, P.; On Highly Composite and Similar Numbers, Transactions of the American Mathematical Society, Vol. 56, No. 3, (November 1944), pp. 448-469.
Nicolas, Jean-Louis; Ordre maximal d'un element du groupe Sn des permutations et "highly composite numbers". Bull. Soc. Math. France 97: (1969), pp. 129-191.
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LINKS
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Eric Weisstein's World of Mathematics, Superabundant Number.
Wikepedia, Highly Abundant Numbers.
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FORMULA
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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).
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EXAMPLE
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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.
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A002093, A004394, A000203, A128700, A128701, A128702.
Sequence in context: A083246 A023492 A022801 this_sequence A104816 A020488 A064435
Adjacent sequences: A128696 A128697 A128698 this_sequence A128700 A128701 A128702
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KEYWORD
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nonn
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AUTHOR
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Ant King (mathstutoring(AT)ntlworld.com), Mar 28 2007
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