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A127661
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Lengths of the infinitary aliquot sequences.
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6
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2, 3, 3, 3, 3, 1, 3, 4, 3, 5, 3, 5, 3, 6, 4, 3, 3, 6, 3, 6, 4, 7, 3, 8, 3, 4, 4, 6, 3, 6, 3, 4, 5, 7, 4, 7, 3, 8, 4, 8, 3, 5, 3, 4, 5, 5, 3, 7, 3, 7, 5, 7, 3, 4, 4, 6, 4, 5, 3, 1, 3, 8, 4, 5, 4, 3, 3, 8, 5, 10, 3, 3, 3, 9, 4, 9, 4, 2, 3, 8, 3, 5, 3, 10, 4, 6, 6, 8, 3, 1, 5, 7, 5, 8, 4, 9, 3, 8, 5, 7
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| The length of an infinitary aliquot sequence is defined to be the length of its transient part + the length of its terminal cycle
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REFERENCES
| Cohen, Graeme L.; On an Integer's Infinitary Divisors, Mathematics of Computation, Vol. 54, No. 189. (1990), pp. 395-411.
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LINKS
| Pedersen, Jan Munch, Tables of Aliquot Cycles.
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EXAMPLE
| a(4)=3 because the infinitary aliquot sequence generated by 4 is <4,1,0> and it has length 3.
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MATHEMATICA
| 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; g[n_] := If[n > 0, properinfinitarydivisorsum[n], 0]; iTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Length[iTrajectory[ # ]] &/@ Range[100]
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CROSSREFS
| Cf. A126168, A127662, A127663, A127664, A127665, A127666, A127667.
Sequence in context: A204916 A110049 A097032 * A008968 A162499 A135715
Adjacent sequences: A127658 A127659 A127660 * A127662 A127663 A127664
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KEYWORD
| nonn
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AUTHOR
| Ant King (mathstutoring(AT)ntlworld.com), Jan 26 2007
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