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



a(n) 

 + (r + 3 2 ){<7(>-2) + <7(n-4)} 

 + (l 2 + 3 2 + 5 2 ) [a (n - 6) + (n -8) + (n- 10)} 

 + (l 2 + 3 2 + 5 2 + 7 2 ){o>-12) + } 



Both series are to be continued as far as the terms remain 

 positive. As an example, putting n = 9, we have 



9a- (9) 

 + 2 {7o- (7) + 5(7 (5)} 

 + 8{8<r(8) + «r(l)}j 



<r(9) 



+ 10{o-(7) + o-(5)} 



+ 35{ -(3) + o-(l)}; 

 that is, 



9 x 13 + 2 {56 + 30} + 3 {12 + 1} = 13 + 10 {8 + 6} + 35 {4 + 1}. 



§ 8. If we denote by £(n) the excess of the sum of the uneven 

 divisors of n over the sum of the even divisors, it is evident that, 

 if n be uneven, £(n) = a- (n); and it can be shewn that if n be even, 

 and = 2 p r, where r is uneven, then 



£(w) = -(2* +1 -3)o-(r). 



This formula is true also when p = and includes the case of 

 n uneven. It is evident that when n is even £ (n) is negative. 



The function £ satisfies a relation of the same kind as Euler's 

 formula, viz. if n be any number, we have 



£0) + £(n - 1) + £0 - 3) + f (n - 6) + £(w - 10) +...= 0, or w, 



according as n is not, or is, a triangular number. 



If therefore we define %(n — n) = £(0) to denote — w, we have, 

 for all values of n, 



C(») + C(n-l) + C(»-3) + C(n-6) + C(»-lQ)+.,. = 0. 



§ 9. The function £(%) may be expressed in terms of the £"s 

 of all the numbers inferior to n by means of the following formula, 

 which differs from the cr-formula in § 5 only in the signs of the 

 terms, which are all positive. 



