|
%I #21 Sep 23 2022 11:39:59
%S 1,2,3,3,5,6,7,7,3,7,11,3,13,7,3,3,17,11,19,7,11,7,23,17,6,3,13,28,29,
%T 3,31,31,3,7,13,17,37,7,17,43,41,3,43,43,3,3,47,41,7,43,11,3,53,3,17,
%U 41,23,31,59,43,61,7,41,41,19,3,67,31,13,43,71,3,73,43,7,41,19,3,79,41,43,43
%N Termination of the aliquot sequence starting at n.
%C Catalan's conjecture [not yet established and probably false] is that every aliquot sequence terminates in a prime number followed by 1, a perfect number, a friendly pair or an aliquot cycle.
%C a(n) = the prime number if the sequence terminates in a prime followed by 1, a(n) = a perfect number if the sequence terminates in a perfect number, a(n) = the smallest number of the cycle if the sequence terminates in an aliquot cycle, a(n) = 0 if the sequence is open ended. So far 276 is the smallest number for which the termination of the aliquot sequence is not known.
%H W. Creyaufmueller, <a href="http://www.aliquot.de/aliquote.htm">Aliquot Sequences</a>.
%H R. J. Mathar, <a href="/A115350/a115350.txt">Table of n, a(n) for n= 1,...,12572</a> with -1 substituted for a(n) where terminations are not yet known.
%H R. J. Mathar, <a href="/A115350/a115350.pdf">Illustration of Aliquot Sequence Mergers</a> (2014)
%H Paul Zimmerman, <a href="http://www.loria.fr/~zimmerma/records/aliquot.html">Aliquot Sequences</a>.
%e a(12)=3 since the aliquot sequence starting at 12 is: 12 - 16 - 15 - 9 - 4 - 3.
%e a(95)=6 since the aliquot sequence starting at 95 is: 95 - 25 - 6 - 6 ...
%t a[n_] := If[n == 1, 1, FixedPointList[If[# > 0, DivisorSigma[1, #] - #, 0]&, n] /. {k__, 1, 0, 0} :> {k} // Last];
%t Array[a, 100] (* _Jean-François Alcover_, Mar 28 2020 *)
%Y Cf. A098007, A098008, A098009, A098010, A003023, A044050, A007906, A037020, A063769, A005114.
%K nonn
%O 1,2
%A _Sergio Pimentel_, Mar 07 2006
%E Edited by _N. J. A. Sloane_, Aug 14 2006
%E a(61)-a(80) from _R. J. Mathar_'s list by _Robert Price_, Mar 16 2019
|
