108 Mr Glaisher, On the sum of the divisors of a number. [Feb. 25, 



February 25, 1884. 

 Mr Glaisher, President, in the chair. 



The following were elected Fellows of the Society:" 



A. E. Forsyth, B.A., Trinity College. 

 W. J. Ibbetson, B.A., Clare College. 



The following communications were made to the Society: 



(1) On the sum of the divisors of a number. By J. W. L. 

 Glaisher, M.A., F.E.S. 



§ 1. Denoting by o- (n) the sum of the divisors of n, it was 

 proved by Euler that 



a (n) - a (n - 1) - a (n - 2) + a (n - 5) + o- (n - 7) - . . . = 0, 



where 1, 2, 5, 7,... are the pentagonal numbers given by the 

 formula \r (3r + 1). The series is to be continued until the 

 arguments become negative and the term a (n — n) or a (0), when 

 it occurs, is to be replaced by n. 



The term <r (n — n) occurs only when n is itself a pentagonal 

 number, and if we make no convention with regard to the meaning 

 to be assigned to a (0), but suppose it to have its proper value 

 zero, the theorem becomes 



o- (n) - o- (n - 1) - a-(n — 2) + cr (n — 5) + a- (n - 7) — . . . 



= or (- ly- 1 n, 



according as n is not, or is, a pentagonal number \r (3r + 1). 



This is the form in which the equation arises as the result 

 of the process by which Euler obtained it. In the second case 

 when n = ^r(Sr±l) the expression on the left-hand side of the 

 equation contains the term (— l) r cr(0), and by conventionally 

 defining cr(n — ri), when it occurs, to denote n, we obtain the 

 theorem in the first form, which is that in which Euler preferred 

 to enunciate it. This theorem was the first one of its kind 

 discovered, and it appeared to Euler to be of the very highest 

 interest, as it afforded a method of calculating the sum of the 

 divisors of a number (and thus also a means of deciding whether 

 it was prime or not) by the sole aid of operations which have no 

 relation whatever to the divisors themselves*. 



§ 2. Euler deduced his formula from the equation 

 (I - cc) (I - cc 2 ) (I - x 5 ) .., =l-x-a;' 2 + a; 5 +x' -x 12 -x 15 +&c. 



* "Observatio de summis divisoruru," Opera Minora Collecta, Vol. i. pp. 146 — 

 154. 



