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Search: id:A129656
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| A129656 |
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Infinitary abundant numbers: integers for which A126168 (n)>n, or equivalently for which A049417 (n)>2n. |
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+0
2
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| 24, 30, 40, 42, 54, 56, 66, 70, 72, 78, 88, 96, 102, 104, 114, 120, 138, 150, 168, 174, 186, 210, 216, 222, 246, 258, 264, 270, 280, 282, 294, 312, 318, 330, 354, 360, 366, 378, 384, 390, 402, 408, 420, 426, 438, 440, 456, 462, 474, 480, 486, 498
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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For large n, the distribution of a(n) is approximately linear and asymptotically satisfies a(n)~7.95n. It follows that the density of the infinitary abundant numbers is 1/7.95, which is about 0.126.
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REFERENCES
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Cohen, Graeme L.; On an Integer's Infinitary Divisors, Mathematics of Computation, Vol. 54, No. 189, (Jan., 1990), pp. 395-411.
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LINKS
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Eric Weisstein's World of Mathematics, definition of infinitary divisor.
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EXAMPLE
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The third integer that is exceeded by its proper infinitary divisor sum is 40. Hence a(3)=40.
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MATHEMATICA
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ExponentList[n_Integer, factors_List]:={#, IntegerExponent[n, # ]}&/@factors; InfinitaryDivisors[1]:={1}; InfinitaryDivisors[n_Integer?Positive]:=Module[ { factors=First/@FactorInteger[n], d=Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f, g}, BitOr[f, g]==g][ #, Last[ # ]]]&/@ Transpose[Last/@ExponentList[ #, factors]&/@d]], _?(And@@#&), {1}]] ]] ] Null; properinfinitarydivisorsum[k_]:=Plus@@InfinitaryDivisors[k]-k; InfinitaryAbundantNumberQ[k_]:=If[properinfinitarydivisorsum[k]>k, True, False]; Select[Range[500], InfinitaryAbundantNumberQ[ # ] &]
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CROSSREFS
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Cf. A126168, A049417, A127666, A129657, A007357..
Sequence in context: A098030 A068544 A109797 this_sequence A048945 A111398 A030626
Adjacent sequences: A129653 A129654 A129655 this_sequence A129657 A129658 A129659
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KEYWORD
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easy,nonn
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AUTHOR
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Ant King (mathstutoring(AT)ntlworld.com), Apr 29 2007
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