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



The formula (i) affords a verification of all values of a(ri) with 

 uneven arguments up to any limit and is very complete ; but the 

 multiplications required in the calculation of the terms are labori- 

 ous. I used this formula to verify the portion from n = 1 to n = 500 

 of the table of a(n) communicated to the Society on January 28. 

 The other three formulae* are in no respect preferable to Euler's 

 formula. 



§ 5. Since the Meeting in January I have obtained the fol- 

 lowing curious formula which serves to express a (n) in terms of the 

 cr's of all the numbers inferior to n, and thus affords a perfect 

 and easy verification of a table of a(n). 



If n be any number, then 



a(n) 



-2a(n-l)-2a(n-2) 



+ So- (n - 3) + Scr (n - 4) + So- (n - 5) 



- 4(r (n - 6) - 4o- (n - 7) - 4o- (n - 8) - 4a (n - 9) 



+ 5a(n-10) + 



-K-irvo-a) 



= (-1)' i(s*-s), 



where s denotes what the coefficient of o-(O) would be if the series 

 were continued one term further. Thus s = r unless the term 

 (— l)''~Vo- (1) is the last of the group for which the coefficient is r, 

 and when this is the case, s = r + 1. 



The expression on the right-hand side of the equation 



= (-l)«i( 5 -l) s ( s +l) 



and is thus obviously an integer. It will be noticed that the 

 value of the series is the same for r consecutive positions of the 

 last term, i.e. the value of the series is the same if the term in- 

 volving o-(l) is the last of the group having r — 1 as coefficient or 

 is any one except the last of the terms having r as coefficient. 



For example, putting n — 5, 6 and 7, we have 



o-(5) 



-2{o-(4) + o-(3)} 



+ 3{<7(2) + o-(l)} = (_l)3i( 3 3-3), 



* The formulas (i) and (ii) were published in the Quarterly Journal of Mathe- 

 matics, Vol. xix. pp. 216, 222 (June, 1883). Since this paper was read the proofs of 

 (iii) and (iv) have been published in the Quarterly Journal, Vol. xx. p. 118. 



