Odd perfect numbers not divisible by 3. II

Abstract

Abstract. We prove that odd perfect numbers not divisible by 3 have at least eleven distinct prime factors.1. N is called a perfect number if a(7V) = 27V, where a(TV) is the sum of positive divisors of TV. Twenty-seven even perfect numbers are known; however, no odd perfect (OP) numbers have been found.Suppose TV is OP and <o(TV) is the number of distinct prime factors of N.Gradstein (1925), Kühnel (1949), Weber (1951, and the author (1978, [5]) proved that co(TV) > 6. Pomerance (1972, [7]) and Robbins (1972) proved that «(TV) > 1. Hagis (1975, [2]) and Chein (1978, [1]) proved that <o(/V) » 8. Hagis and McDaniel [3] proved that the largest prime factor of TV > 100129, and Pomerance [8] proved that the second largest prime factor of TV > 139.If 31 TV, then Kanold (1949) proved that co(TV) s= 9, and the author (1977, [4]) proved that to(TV) > 10.In this paper we prove Theorem. If N is OP and3\N, then <o(TV) > 11.2. In the remainder of this paper we assume that TV is OP and

On some properties of the Euler's factor of certain odd perfect numbers

“…We note that the number of distinct prime factors (n) of an odd perfect number n is (n) 8 [2] (if 3 does not divide n, then (n) 11, [3,4]). A recent result of this kind is: if (n) = 8, then 5|n [8].…”

A rationality condition for the existence of odd perfect numbers

“…, is bounded above by 8.032 × 10 −5 when there are eight different prime factors [2] and 1.004×10 −6 when there are eleven different prime factors not including 3 [13] [14]. Given that the probability of an integer being a square is independent of it being expressible in terms of a product of repunits, the density of square-full numbers having the form 2(4k + 1) σ (4k + 1)…”