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IV. On the Connexion between Recurring Formula: involving 

 Sums of Divisors and the Corresponding Formula involving 

 Differences between Sums of Even and Uneven Divisors. By 

 J. W. L. Glaisher, Sc.D., F.R.S.* 



§ 1. (~\F all the numerous recurring formulae relating to 

 ^-^ the divisors of numbers perhaps the simplest is 



rW+?(^-l) + f(n-3) + ^-6) + ?(72-10)+&c. = 0t, 



where ? (n) denotes the excess of the sum of the uneven 

 divisors of n above the sum of the even divisors (this excess 

 being of course negative when n is even) ; and 5(0), which 

 occurs when n is a triangular number, is to have the conven- 

 tional value — n. 



This theorem asserts that, n being any number, the sum of 

 the uneven divisors of n, n — 1, n — 3, &c. is equal to the sum 

 of the even divisors of these same numbers, if n is not a tri- 

 angular number ; and that when n is a triangular number, 

 the former sum exceeds the latter sum by n. 



For example, let n=d, which is not a triangular number. 

 The uneven divisors of 9, 8, 6, 3 are 1, 3, 9 ; 1 ; 1, 3 ; 1,3, 

 the sum of which is 22. The even divisors of the same 

 numbers are 2, 4, 8 ; 2, 6, the sum of which is also 22. 



As another example let n=10, which is a triangular num- 

 ber. The uneven divisors of 10, 9, 7, 4 are 1,5; 1, 3, 9 ; 

 1,7; 1, the sum of which is 28. The even divisors are 

 2,10; 2,4, the sum of which is 18. The excess of the 

 former sum over the latter sum is therefore equal to n. 



§ 2. The corresponding formula (i. e. the formula having 

 the same arguments) which relates to the sums of divisors is 

 o-(n)-3o-(7i-l)+5<r(n-3)-7o-(7i-6)+9<7(w-10)-&c. = 0t, 

 where <r(n) denotes the sum of all the divisors of n, and cr(0) 

 when it occurs, is to have the value ^n. 



§ 3. The object of this note is to point out that the two 

 formulae are derivable from a single general theorem relating 

 to the actual divisors of the numbers n, n — 1, n — 3, &c. It 

 is especially interesting to notice how it comes about that the 

 coefficients are 1, —3, 5, —7, &c. in the one case and that they 

 are all unity in the other. 



§ 4. The general theorem may be stated as follows : — Let 

 G r {(p(d), yfr(d), . . . } denote the group of numbers 



* Communicated by the Author. 



t Proc. Lond. Math. Soc. vol. xv. (1884) p. 110: or Proc. Camb. Phil. 

 Soc. vol. v. (1884) p. 116. 



I Quart. Journ. Math. vol. xix. Q883) p. 220 : or Proc. Camb. Phil. 

 Soc. vol. v. (1884) p. 109. 



