Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

If the half of each term of this series is taken, so as to produce
the series + + * + & f & + &, kc. then the half of the
sum of the denominator of the last term, and the denominator
of the term immediately preceding the last, multiplied by the
last term, will be equal to the sum of the series. If the 3rd of
each term is taken, the sum of the two denominators must be
divided by 3. If the 4th of each term is taken, the division
must be by 4; and so of the rest.
If the half of each term of the series 1 + fr + f + + &, kc.
+ f + + & + &, kc.
is taken, so as to produce the series
and if the sum of the denominators is divided by 2, the quotient
multiplied by the last term is equal to the sum of the ser-
2+4+8+16 XL-A
ies. ~hus
- 2 16 - 2
2+4+8 ~t-.L+++i,
+ f + + &, and so of the rest.
But if the third of each term of this series is taken, the divisor
must be 3; if the 4th is taken, the divisor must be 4; if
the 5th, is must be 5; if the 6th, 6; and so on.
CHAPTER XXXVI
On imperfectly amicable numbers.
IMPERFECIZY amicable numbers are such as 27, and 35, 39
and 55,65 and 77,51 and 91,95 and 119,69 and 133, 115 and
187, 87 and 247. For the parts of 2733x9 are 1, 3, 9, the
sum of which is 13; and the parts of 35=5)