Chap. X] SUM AND NuMBER OF DiVISORS. 311 
He proved two theorems relating to the divisors of 1, 2, . . ., w: 
+ (G^n-3 + ^n-4 + G^n-5)( _y_3l ) — • • • 
all cancel with the exception of —2, — 4, . . ., — (p— 2), each taken twice, 
p taken p — 1 times and — 0, if p be even; but with the exception of 1, 3, ... , 
p — 2, each taken twice, and —p taken p — 1 times, if p be odd, where 
p(p+l)/2 is the triangular number next >n; 
all cancel with the exception of k taken k times, for A; = 1, 3, 5, . . . , p — 1, if p 
be even; and of —A; taken k times, for A; = 2, 4, 6, . . . , p — 1, if p be odd; here 
zeros are ignored. 
The last two theorems yield (as before) corresponding relations for any- 
even function x aiid any odd function xp. Applying them to x{d+l) 
= (d+l)"' and \l/{d)=d"*, where m is odd, and in the first case dividing 
by 2(m+l), and modifying the right members, we get for 
r=o-Jn)-2{(T^(n-l)+(r^(n-2)}+3{Mn-3) 
+(rm{n-4:)+(Tm{n-5)]-. .. 
the respective relations 
+3*+^ (next three) - . . . } 
-^-^M2(m^"^^^ +3 V2J^P -5V4/3"^ +...=^2 -p^, 
where s = (m+l)/2 and 0-^(0) terms are suppressed; 
!r=S2(^^{(r^_,(n-l)+(7^_fc(n-2)-2^ (next three) + ( 1^3') (next four) 
_(2^+4*^) (next five) + (lH3*+5'=) (next six) - (2H4H6') (next seven) + . . . } 
r ^m+i_^^m+i_^^m+i_^ _{_ (^ _ l)-+i^ if p bg eveu, 
i-|_ 2^+1 _ 4^+1 _gm+i_ _(p_i)m+i^ if p be odd, 
where, in each, A: takes the values 2, 4, . . . , m — 1. These sums of like powers 
of odd or even numbers are expressed by the same function of Bernoullian 
numbers. For m = l, the first formula becomes that by Glaisher,^^ repub- 
Ushed.^^ Three further (t„ formulas are given, but not applied to o-„. 
J. Hammond"^ wrote (n; m) = l or according as n/m is integral or 
fractional, also t{x) =a{x)=0 if x is fractional, and stated that 
00 00 
T(n/m) = S (n; jm), a (n/m) = 2 j(n; jm). 
y=i }=i 
"^Messenger Math., 20, 1890-1, 158-163. 
