NECESSARY CONDITIONS FOR EXISTENCE OF A FRIEND OF 38

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 ����.