50 History of the Theory of Numbers. [Chap, i 
P. Poulet'^' discovered the chain of period five, 
71 = 12496 = 24-11.71, s(n)= 2^- 1947, s\n) = 2^-967, s\n) =2^-23-79, 
sHn) =2^.1783, 
with s^{n) =n; and noted that 14316 leads a chain of 28 terms. 
Generalizations of Amicable Numbers. 
Daniel Schwenter^^ noted in 1636 that 27 and 35 have the same sum of 
ahquot parts. Kraft^^^ noted in 1749 that this is true of the pairs 45, 3-29; 
39, 55; 93, 145; and 45, 13-19. In 1823, Thomas Taylor^^^ called two such 
numbers imperfectly amicable, citing the pairs 27, 35; 39, 55; 65, 77; 51, 
91; 95, 119; 69, 133; 115, 187; 87, 247. George Peacock^°o used the same 
term. 
E. B. Escott^"^ asked if there exist three or more numbers such that each 
equals the sum of the [aliquot] divisors of the others. 
A. G^rardin^°^ called numbers with the same sum of aliquot parts 
nombres associes, citing 6 and 25; 5-19, 7-17, and 11-13, and many more sets. 
An equivalent definition is that the n numbers be such that the product of 
n — 1 by the sum of the aliquot divisors of any one of them shall equal the 
sum of the aliquot divisors of the remaining n — 1 numbers. 
L. E. Dickson^°^ defined an amicable triple to be three numbers such 
that the sum of the aliquot divisors of each equals the sum of the remaining 
two numbers. After developing a theory analogous to that by Euler^®* for 
amicable numbers, Dickson obtained eight sets of amicable triples in which 
two of the numbers are equal, and two triples of distinct numbers: 
293-3370, 5- 16561a, 99371o (a = 25.3-13), 
3-896, 11-296, 3596 (6 = 2i*.5-19-31-151). 
^L'intermddiaire des math., 25. 1918, lOO-l. 
♦""Encyclopaedia Metropolitana, London, I, 1845, 422. 
«"L'interm6diaire des math., 6, 1899, 152. 
♦""Sphinx-Oedipe, 1907-8, 81-83. 
♦MAmer. Math. Monthly, 20, 1913, 84-92. 
