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



D(n) 



-2A(n-l) + 2A (n - 2) 



+ 3D (n - 3) + 3D (n - 4) + 3D (n - 5) 



-4A(w-6)- 



= 0. 



The argument is to be included in both formulae, and we define 

 A (0) to denote and sD (0) to denote s . I (s 2 - 1) = £ (s 3 - s). 



For example, putting n = 5 and 6, we have 



A (5) 



- 2D (4) - 2D (3) 



+ 3A (2) + 3A (1) + 3A (0) = 0, 

 that is, 6-12-0 + 3 + 3 + = 0; 



D(5) 



- 2A (4) - 2A (3) 



+ SD (2) + 3D (1) + SD (0) = 0, 

 that is, - 2 - 8 + 6 + + i (3 3 - 3) = ; 



A (6) 



- 2D (5) - 2D (4) 



+ 3A (3) + 3A (2) + 3A (1) 

 -4Z>(0) =0, 



that is, 4 - - 12 + 12 + 3 + 3 - i (4 3 - 4) = ; 



D(6) 



- 2A (5) - 2A (4) 



+ 3Z>(3) + 3Z>(2) + 3Z>(1) 

 -4A(0) =0, 



that is, 8-12-2 + + 6 + 0-0 = 0. 



§ 11. The function which expresses the excess of the number 

 of divisors of n which have the form 4m 4 1 over the number of 

 divisors which have the form 4m + 3 satisfies a relation so similar 

 to the o--formula of § 5 and the ^-formula of § 9 as to be deserving 

 of notice in connection with these two formulae. 



Denoting this function by E(n), we find that, n being un- 

 restricted, 



