Recursive determination of the sum-of-divisors function
نویسندگان
چکیده
منابع مشابه
Iterating the Sum-of-Divisors Function
1991 Mathematics Subject Classi cation: 11A25, 11Y70 Let 0(n) = n and m(n) = ( m 1(n)), where m 1 and is the sum-of-divisors function. We say that n is (m; k)perfect if m(n) = kn. We have tabulated all (2; k)-perfect numbers up to 109 and all (3; k)and (4; k)-perfect numbers up to 2 108. These tables have suggested several conjectures, some of which we prove here. We ask in particular: For any ...
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15 صفحه اولThe range of the sum-of-proper-divisors function
Answering a question of Erdős, we show that a positive proportion of even numbers are in the form s(n), where s(n) = σ(n) − n, the sum of proper divisors of n.
متن کاملRemarks on fibers of the sum-of-divisors function
Let σ denote the usual sum-of-divisors function. We show that every positive real number can be approximated arbitrarily closely by a fraction m/n with σ(m) = σ(n). This answers in the affirmative a question of Erdős. We also show that for almost all of the elements v of σ(N), the members of the fiber σ−1(v) all share the same largest prime factor. We describe an application of the second resul...
متن کاملSome Problems of Erdős on the Sum-of-divisors Function
Let σ(n) denote the sum of all of the positive divisors of n, and let s(n) = σ(n)− n denote the sum of the proper divisors of n. The functions σ(·) and s(·) were favorite subjects of investigation by the late Paul Erdős. Here we revisit three themes from Erdős’s work on these functions. First, we improve the upper and lower bounds for the counting function of numbers n with n deficient but s(n)...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1979
ISSN: 0002-9939
DOI: 10.1090/s0002-9939-1979-0516458-2