A115350 Termination of the aliquot sequence starting at n.

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, 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, 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

Offset: 1

Author

Sergio Pimentel, Mar 07 2006

Keywords

• nonn

Comments

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.

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.

Examples

			a(12)=3 since the aliquot sequence starting at 12 is: 12 - 16 - 15 - 9 - 4 - 3.
a(95)=6 since the aliquot sequence starting at 95 is: 95 - 25 - 6 - 6 ...
		

Links

• W. Creyaufmueller, Aliquot Sequences.
• R. J. Mathar, Table of n, a(n) for n= 1,...,12572 with -1 substituted for a(n) where terminations are not yet known.
• R. J. Mathar, Illustration of Aliquot Sequence Mergers (2014)
• Paul Zimmerman, Aliquot Sequences.

Crossrefs

Cf. A098007, A098008, A098009, A098010, A003023, A044050, A007906, A037020, A063769, A005114.

Programs

Mathematica

a[n_] := If[n == 1, 1, FixedPointList[If[# > 0, DivisorSigma[1, #] - #, 0]&, n] /. {k__, 1, 0, 0} :> {k} // Last];
Array[a, 100] (* Jean-François Alcover, Mar 28 2020 *)

Extensions

Edited by N. J. A. Sloane, Aug 14 2006

a(61)-a(80) from R. J. Mathar's list by Robert Price, Mar 16 2019