Abstract:
This paper sets out to, systematically, use properties of the abundancy index
function to prove that a friend of 38 must be an odd non-square multiple of 19
which is not divisible by 3, and that every prime factor of such that 4|( + 1)
has an even exponent in the prime factorization of . In addition, if the power of
19 in is 2, then 127| in which case the power of 127 must be even, larger
than 2 and not equal to 8, and if the power of 19 is 6, both 701 and 70841 would
be compulsory prime factors of , where the power of 701 cannot equal 1 or 3.
The paper also establishes that it is not possible to have 8 as the power of 19 in the
prime factorization of .
