Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

Farther still, if 2 be subtracted from the number of the rank
which any number after 28, holds in the series 6, 28,496, 8128,
130816, kc. and the remainder be added to the number of the
said rank, the sum will be the index of that power of 2 which
by multiplication with its corresponding number in the series
3, 7, 31, 127, 511, &c. produced the perfect or partially perfect
number. Thus if from 3, we subtract 2, and add 3 to the remainder
1, the sum 4 will indicate that the fourth power of 2,
viz. 16 multiplied by the third term 31, will give the third
term, viz. 4%, in the series 6, 28, 496, 8128, &c. Thus also, if 2
be subtracted from 4, and 4 be added to the remainder, the
sum 6 will indicate that the sixth power of 2, viz. 64, multiplied
by the 4th term 127, will produce the 4th term 8128, of the
series 6, 28, 496, 8128, &c. And so of the rest.
The rule for obtaining a number which is either perfect or
partially perfect in the series 6, 28, 496, 8128, 130816, &c. any
term in this series being given, in the most expeditious manner,
is the following. Multiply the given perfect, or partially
perfect number by 16, and add to the product twelve times
the number in the duple series, from the multiplication of
which with a corresponding number in the series 3, 7,31, 127,
511, &c. the perfect, or partially perfect number is produced,
and the sum will be the next number in order in the series 6,
28, 496, &c. Thus 28 x 16=448, and 448 added to 12 times 4,
i.e. to 48, is equal to 496. Thus too 496X 16=7936, and 7936
+ 12 X 16(=192)=8128. And so of the rest.
Lastly, all tllr pcrfect numbers arc found in the series of
hexagonal numbers; which numbers are 1, 6, 15, 28, 45, 66,
&c.; and the expression which is the aggregate of them and
when evolved gives all of them in order ad infinitum is
1+3