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Search: id:A129657
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| A129657 |
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Infinitary deficient numbers: integers for which A126168 (n)<n, or equivalently for which A049417 (n)<2n. |
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+0
2
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| 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84
(list; graph; listen)
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OFFSET
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1,2
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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)~1.144n. It follows that the density of the infinitary deficient numbers is 1/1.144, which is about 0.874.
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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 sixth integer that exceeds its proper infinitary divisor sum is 7. Hence a(6)=7.
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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; InfinitaryDeficientNumberQ[k_]:=If[properinfinitarydivisorsum[k]<k, True, False]; Select[Range[100], InfinitaryDeficientNumberQ[ # ] &]
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CROSSREFS
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Cf. A126168, A049417, A127666, A129656, A007357.
Sequence in context: A054027 A080907 A127161 this_sequence A103679 A029916 A052414
Adjacent sequences: A129654 A129655 A129656 this_sequence A129658 A129659 A129660
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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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