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A003023
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"Length" of aliquot sequence for n.
(Formerly M0062)
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8
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0, 1, 1, 2, 1, 1, 1, 2, 3, 3, 1, 6, 1, 4, 4, 5, 1, 3, 1, 6, 2, 5, 1, 4, 2, 6, 2, 1, 1, 14, 1, 2, 5, 7, 2, 3, 1, 6, 2, 3, 1, 13, 1, 4, 6, 7, 1, 5, 3, 2, 3, 8, 1, 12, 2, 4, 2, 3, 1, 10, 1, 8, 2, 3, 2, 11, 1, 4, 3, 5, 1, 8, 1, 4, 4, 4, 2, 10, 1, 6, 4, 5, 1, 5, 2, 8, 6, 6, 1, 9, 3, 5, 3, 3, 3, 8, 1, 2, 3, 4, 1, 17
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| The aliquot sequence for n is the trajectory of n under repeated application of the map x -> sigma(x) - x.
The trajectory will either have a transient part followed by a cyclic part, or will have an infinite transient part and never cycle.
Sequence gives (length of transient part of trajectory) - 1 + (length of cycle provided cycle is nonzero). See A098007 for a better version.
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REFERENCES
| G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
R. K. Guy, Unsolved Problems in Number Theory, B6.
R. K. Guy and J. L. Selfridge, Interim report on aliquot series, pp. 557-580 of Proceedings Manitoba Conference on Numerical Mathematics. University of Manitoba, Winnipeg, Oct 1971.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Matthew M. Conroy, Home page (listed instead of email address)
F. Richman, Aliquot series:Abundant,deficient,perfect
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EXAMPLE
| Examples of trajectories:
1, 0, 0, ...
2, 1, 0, 0, ...
3, 1, 0, 0, ... (and similarly for any prime)
4, 3, 1, 0, 0, ...
5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
6, 6, 6, ... (and similarly for any perfect number)
8, 7, 1, 0, 0, ...
9, 4, 3, 1, 0, 0, ...
12, 16, 15, 9, 4, 3, 1, 0, 0, ...
14, 10, 8, 7, 1, 0, 0, ...
25, 6, 6, 6, ...
28, 28, 28, ... (the next perfect number)
30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0, 0, ...
42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0, 0, ...
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MAPLE
| f:=proc(n) local t1, i, j, k; t1:=[n]; for i from 2 to 50 do j:= t1[i-1]; k:=sigma(j)-j; t1:=[op(t1), k]; od: t1; end; # produces trajectory for n
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MATHEMATICA
| f[x_] := (k++; DivisorSigma[1, x] - x); f[1] = 1;
Table[k = 0; FixedPoint[f, n]; k, {n, 1, 102}]
(* From Jean-François Alcover, Jul 27 2011 *)
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PROG
| (MuPAD) s := func(_plus(op(numlib::divisors(n)))-n, n): A003023 := proc(n) local i, T, m; begin m := n; i := 1; while T[ m ]<>1 and m<>1 do T[ m ] := 1; m := s(m); i := i+1 end_while; i-1 end_proc:
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CROSSREFS
| Cf. A098007.
Cf. A059447 (least k such that n is the length of the aliquot sequence for k).
Sequence in context: A039958 A029344 A125769 * A156070 A114731 A035389
Adjacent sequences: A003020 A003021 A003022 * A003024 A003025 A003026
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Matthew Conroy (list1(AT)madandmoonly.com), Jan 16 2006
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