Does Ten Have a Friend?
نویسنده
چکیده
Any positive integer n other than 10 with abundancy index 9/5 must be a square with at least 6 distinct prime factors, the smallest being 5. Further, at least one of the prime factors must be congruent to 1 modulo 3 and appear with an exponent congruent to 2 modulo 6 in the prime power factorization of n. 1 The Abundancy Index For a positive integer n, the sum of the positive divisors of n is denoted σ(n); the ratio σ(n) n is known as the abundancy ratio or abundancy index of n, denoted I(n). A perfect number is a positive integer n satisfying I(n) = 2. Considering the millenia-old interest in perfect numbers and the (at least) centuries-old interest in the “abundancy” of positive integers, it is somewhat surprising that study of the abundancy index seems to have flourished only relatively recently; see [2], [5], and [6], and the references there to earlier work. Interesting questions have been asked and answered: for instance, it is now known ([5] and [6]) that both the range of the function I and the complement of that range in the rational numbers are dense in the interval (1,∞). Questions about another kind of density remain. Let, for x > 1, J(x) = I((x,∞)) = {n | I(n) > x}; does the limit f(x) = lim N→∞ |J(x) ∩ {1, . . . , N}| N (1) exist? If so, what can be said about the behavior of the non-increasing function f? Is it continuous? Strictly decreasing? The open question about the abundancy index to be addressed here, stated in the title and explained in the next section, is not so exotic – in fact, it has a
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تاریخ انتشار 2008