Search: id:A003023
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%I A003023 M0062
%S A003023 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,
%T A003023 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,
%U A003023 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
%N A003023 "Length" of aliquot sequence for n.
%C A003023 The aliquot sequence for n is the trajectory of n under repeated application 
               of the map x -> sigma(x) - x.
%C A003023 The trajectory will either have a transient part followed by a cyclic 
               part, or will have an infinite transient part and never cycle.
%C A003023 Sequence gives (length of transient part of trajectory) - 1 + (length 
               of cycle provided cycle is nonzero). See A098007 for a better version.
%D A003023 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence 
               Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A003023 R. K. Guy, Unsolved Problems in Number Theory, B6.
%D A003023 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.
%D A003023 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A003023 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               AliquotSequence.html">Link to a section of The World of Mathematics.</
               a>
%H A003023 Matthew M. Conroy, <a href="http://www.madandmoonly.com/doctormatt">Home 
               page</a> (listed instead of email address)
%H A003023 F. Richman, <a href="http://www.math.fau.edu/Richman/mla/aliquot.htm">
               Aliquot series:Abundant,deficient,perfect</a>
%e A003023 Examples of trajectories:
%e A003023 1, 0, 0, ...
%e A003023 2, 1, 0, 0, ...
%e A003023 3, 1, 0, 0, ... (and similarly for any prime)
%e A003023 4, 3, 1, 0, 0, ...
%e A003023 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,
%e A003023 6, 6, 6, ... (and similarly for any perfect number)
%e A003023 8, 7, 1, 0, 0, ...
%e A003023 9, 4, 3, 1, 0, 0, ...
%e A003023 12, 16, 15, 9, 4, 3, 1, 0, 0, ...
%e A003023 14, 10, 8, 7, 1, 0, 0, ...
%e A003023 25, 6, 6, 6, ...
%e A003023 28, 28, 28, ... (the next perfect number)
%e A003023 30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0, 0, ...
%e A003023 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0, 0, ...
%p A003023 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
%o A003023 (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:
%Y A003023 Cf. A098007.
%Y A003023 Cf. A059447 (least k such that n is the length of the aliquot sequence 
               for k).
%Y A003023 Sequence in context: A039958 A029344 A125769 this_sequence A156070 A114731 
               A035389
%Y A003023 Adjacent sequences: A003020 A003021 A003022 this_sequence A003024 A003025 
               A003026
%K A003023 nonn,easy
%O A003023 1,4
%A A003023 N. J. A. Sloane (njas(AT)research.att.com).
%E A003023 More terms from Matthew Conroy (list1(AT)madandmoonly.com), Jan 16 2006

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