1884.] Mr Glaisher, On the sum of the divisors of a number. 117 

 If n be any number, then 



too 



+ 2f(n-l) + 2£(n-2) 



+ 3£(w-3) + 3£(rc -4) + 3£(ro - 5) 



+ 4£(?i-6)+ 



+ rr(l) 



where, as in § 5. s = r, unless rcr.(l) is the last term of a group, in 

 which case s = r + l. 



As examples of the formula we find, putting n = 5 and 6, 



?(5) 



+ 2 {£(4) + £(3)} 



+ 3{£(2) + £(l)} = *(3»-3), 



t(6) 



+ 2 {£(5) + £(4)} 



+ 3{£(3) + £(2) + £(l)}=i(4 3 -4); 



that is, 6 + 2 {- 5 + 4} + 3 { - 1 + 1} = 4, 



-4 + 2{ 6 -5} + 3 {4- 1 + 1} = 10. 



If the series be continued one term further so as to include 

 the term s£(0). and if we define £ (0) to denote — ^(s 2 — 1), we 

 may replace the right-hand member of the equation by zero. 



§ 10. If we denote by A (n) the sum of the uneven divisors 

 of n and by D (n) the sum of the even divisors (so that D (n) is 

 zero when n is uneven), then 



a (n) = A (n) + D (n), 



Z(n)=A(n)-D(n), 



and, by addition and subtraction from the formulae in §§ 5 and 9, 

 we find that, n being unrestricted, 



A(n) 



- 2D (n - 1) - 2D (n - 2) 



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



-4D(?i-G)- 



0; 



