A LIST OF ALIQUOT CYCLES OF LENGTH GREATER THAN 2 The number in the first column identifies the cycle; cycles are numbered in increasing order of the smallest member. The numbers in the second column of the file are the numbers occurring in each aliquot cycle; the third column contains the prime factorizations of the numbers in the second column. The aliquot cycles are separated by blank lines. There are 175 cycles in all: 165 of length 4, 1 of length 5, 5 of length 6, 2 of length 8, 1 of length 9, and 1 of length 28. P. Moews and I have exhaustively searched for all aliquot cycles of length greater than 2 such that the number preceding the largest number of the cycle is below 1.54*10^11. As of November 1997, Jan Otto Munch Pedersen had carried this search up to 4.27*10^11; as of January 2001, he had carried it up to 4.5*10^11; as of February 2003, up to 6*10^11; and as of July 2004, up to 6.1*10^11. As of November 2005, Andre Needham had carried this search up to 10^12; as of May 2006, up to 2*10^12; as of September 2006, up to 3*10^12; as of November 2006, up to 4*10^12; and as of September 2007, up to 5*10^12. All cycles found are among those shown below. Some of the cycles of length 4 are of a form given by Borho (see the reference below) analogous to Euler's first form for amicable pairs. These cycles are numbers 10, 22, 36, 40, 45, 56, 58, and 61-175. Credits: (If two or more people seem to have independently discovered the same cycle, it will be credited to both or all of them.) 1,2: P. Poulet, #4865, L'intermediare des math. 25 (1918), pp. 100-101. 3-11: H. Cohen, On amicable and sociable numbers, Math. Comp. 24 (1970), pp. 423-429. 3-9,11-16: S. C. Root, Item 61, MIT AI memo 239 (HAKMEM), Feb. 29, 1972. 10: W. Borho, Ueber die Fixpunkte der k-fach iterierten Teilersummenfunktion, Mitt. Math. Gesellsch. Hamburg 9 (1969), pp. 34-48. 12-15: Richard David, letter to D. H. Lehmer, 72:02:25; cf. J. S. Devitt, R. K. Guy, and J. L. Selfridge, Third report on aliquot sequences, Congr. Numer. XVIII, Proc. 6th Manitoba Conf. Numerical Math. 1976, pp. 177-204. 17-27: A. Flammenkamp, New sociable numbers, Math. Comp. 56 (1991), pp. 871-873. 18,19,21-29,36,45,72, 80,112,117: D. Moews and P. C. Moews, A search for aliquot cycles below 10^10, Math. Comp. 57 (1991), pp. 849-855. 18-24,28,29,31,36,37,40: R. A. Brodie (unpublished.) 22,36,64,69,71,75,78,80, 82,83,85,87,91,93, 95-97,99,100,104,108, 113,114,118,119,121, 123,129,132,135, 139-141,145,150,153, 157,158,168,175: Ren Yuanhua (unpublished?) 30-33: D. Moews and P. C. Moews, A search for aliquot cycles and amicable pairs, Math. Comp. 61 (1993), pp. 935-938. 34,35,37-40,73,98,101: D. Moews and P. C. Moews (unpublished.) 41,52,58,61-63,65,68,79, 81,90,109,131,134,142: J. O. M. Pedersen (as above; unpublished.) 42,46,48-51,53,54: A. Needham 43,47: H. Baader (unpublished?) 44,94: M. Marcus (unpublished?) 55,57,59,60: Dmitry Bodyagin (unpublished?) 56,66,67,70,74,76,77,84, 86,88,89,92,102,103, 105,107,110,111,115, 116,120,122,124-128, 130,133,136-138,143, 144,146,148,149,151, 152,154-156,159-167, 169-174: Karsten Blankenagel & W. Borho. (many of these appear in Blankenagel's thesis: "Befreundete Vierzyklen", Diplomarbeit, Jun 2001, Mathematik der Bergishen Universitaet - Gesamthochschule Wuppertal.) 106,147: A. vom Stein, K. Blankenagel, W. Borho. (cf. Blankenagel reference above.) Also, apart from cycles 88, 148, 154, 156, 162-164, 170, and 174, the cycles found by Blankenagel and Borho and by vom Stein, Blankenagel, and Borho can all be found in: New amicable four-cycles, Math. Comp. 72 (2003), pp. 2071-2076. 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