On characterizing potential friends of 20

Does 20 have a friend? Or is it a solitary number? A folklore conjecture asserts that 20 has no friends, i.e., it is a solitary number. In this article, we prove that a friend N of 20 is of the form N = 2 · 5^2a ·m², with (3;m) = (7;m) = 1 and it has at least six distinct prime divisors. Furthermore, we show that Ω (N) ≥ 2ω (N) + 6a − 5 and if Ω (m) ≤ K then N < 10 · 6^{(2K−2a+3−1)2}, where Ω(n) and ω(n) denote the total number of prime divisors and the number of distinct prime divisors of the integer n respectively. In addition, we deduce that not all exponents of odd prime divisors of friend N of 20 are congruent to −1 modulo f, where f is the order of 5 in (ℤ/pℤ)^× such that 3 | f and p is a prime congruent to 1 modulo 6. Also, we prove necessary upper bounds for all prime divisors of friends of 20 in terms of the number of divisors of the friend. In addition, we prove that if P is the largest prime divisor of N, then P<N14 P < {N^{{1 \over 4}}} .