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



If P denote the expression 



a (n) 



- a(n - 2) - a- (n - 3) - tr (n - 4) 



+ a(n - 7) + <r(n - 8) + a(n - 9) + crin - 10) + <r(w - 11) 



-0-0-15)- 



where between the groups of one, three, five, seven, ... terms 

 having alternate signs there are gaps of one, two, three, . . . terms, 

 and the series is to be continued as long as the arguments remain 

 positive, the term o- (0) being included when it occurs but having 

 the value zero ; then, if a (0) occurs as a term, 



P=(-ir 1 %r(r+l), 



where r denotes the number of terms omitted in the gap preced- 

 ing the group of terms in which a (0) occurs, and, if a (0) does 

 not occur as a term, 



p = (-l)'(r + l), 



where r denotes the number of terms omitted in the last gap in 

 the series. 



For example, putting n = 7, we have 



o-(7) 

 - o-(5) - o-(4) - o-(3) 

 + <r(0) =-fx2x3, 



that is, 8-6-7-4+0=- 9. 



In this case r, the number of terms omitted between a (3) and 

 <r(0),=2. 



Putting n = 8, we have 



<r(8) 

 -o-(6)-o-(5)-o-(4) 

 + o-(l) + o-(0) =-fx2x3, 



that is, 15 -12- 6 -7 + 1 =-9, 



r being equal to 2 as before. 



Putting n = 6, we have 



a (6) 



-o-(4)-o-(3)-o-(2) =-2, 



that is, 12 - 7 - 4 - 3 = - 2, 



for here, r, the number of terms omitted between a (6) and a (4),=1. 



