A080907 Numbers whose aliquot sequence terminates in a 1.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Author

Gabriel Cunningham (gcasey(AT)mit.edu), Mar 31 2003

Keywords

• nonn
• nice

Comments

All primes are in this set because s(p) = 1 for p prime. Perfect numbers are clearly not in this set. Neither are aspiring numbers (A063769), or numbers whose aliquot sequence is a cycle (such as 220 and 284).

There are some numbers whose aliquot sequences haven't been fully determined (such as 276).

Examples

			4 is in this set because its aliquot chain is 4->3->1. 6 is not in this set because it is perfect. 25 is not in this set because its aliquot chain is 25->6.
		

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..1000
• M. Benito, W. Creyaufmüller, J. L. Varona, and P. Zimmermann, Aliquot Sequence 3630 Ends After Reaching 100 Digits, Experimental Mathematics (2002), Volume 11, issue 2.
• Markus Tervooren, Aliquot sequence of 840
• Eric Weisstein's World of Mathematics, Aliquot Sequence

Crossrefs

Cf. A063769, A115350, A131884.

Complement of A126016.

Programs

Mathematica

maxAliquot = 10^50; A131884 = {}; s[1] = 1; s[n_] := DivisorSigma[1, n] - n; selQ[n_ /; n <= 5] = True; selQ[n_] := NestWhile[s, n, If[{##}[[-1]] > maxAliquot, Print["A131884: ", n]; AppendTo[A131884, n]; False, Length[{##}] < 4 || {##}[[-4 ;; -3]] != {##}[[-2 ;; -1]]] & , All] == 1; Select[Range[1, 1100], selQ] (* Jean-François Alcover, Nov 14 2013, updated Sep 10 2015 *)

Formula

n is a member if n = 1 or s(n) is a member, where s(n) is the sum of the proper factors of n.

Extensions

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

More terms from Franklin T. Adams-Watters, Dec 14 2006

The fact that 840 was missing from the sequence b-file was pointed out by Philip Turecek, Sep 10 2015