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XXIV. Some Properties of the Sum of the Divfors of Numbers . 
By Edward Waring, M. D. P. R, S’. 
Read May 29, 1788. 
- u: 
ET the equation x — 1 . x 2 - 1 . x 3 - 1 . x - 1 . x i - 1 
x n — 1 — x h — px b ~ l 4 qx h ~~ z — rx h ~> 4- sx b ~* — &c. = 
x b - a **" 1 — x‘ ~ l 4 x L ~s~+-x h ~ 7 — x h ~ lz — x b ~ 4 a ' :, “' 22 4 A^’ -26 — 
a a ~ 3 5 _ % b — 40 4 a 6-5 * 4 57 — &c. * . . a , j— " &c. = A = o. The 
figns q- and — proceed alternately by pairs unto the term 
x b ~ n . The co-efficients of all the terms to the above men- 
tioned (x b ~ ) will be4», — 1 or o ; they will be4i, when 
multiplied into x b ~“ v i where 1; = — — - or = - , and 2 
an 
2 2 
even number ; but — 1, if 3 be an uneven number ; in all other 
cales they will be = o. 
The numbers 1, 2, 5, 7, 12, 15, 22, 26, 39, 40, &c. 
fubt rafted from h may be collefted from the addition of the 
numbers 1, 1, 3, 2, 5, 3, 7, 4, 9, 5, ir, 6, See. which 
confift of two arithmetical feriefes 1, 3, 5, 7, 9, 11, &c« 
], 2, 3, 4, 5, 6, 7, &c. intermixed. 
2. The fum of any power (w) of each of the roots in the 
equation A = o will be S (w), where S(w) denotes the fum of 
ail the divifors of the number m, if m be not greater than n. 
Cor . Hence (by the rule for finding the fum of (/») powers 
of each of the roots from the fum of the inferior powers and 
co-efficients of the given equation) may be deduced S(w) = 
7 /s 
