Multiperfect numbers with identical digits

Perfect Numbers with Identical Digits

Suppose g ≥ 2. A natural number N is called a repdigit in base g if it has the shape a g −1 g−1 for some 1 ≤ a < g, i.e., if all of its digits in its base g expansion are equal. The number N is called perfect if σ(N) = 2N , where σ(N) := � d|N d is the usual sum of divisors function. We show that in each base g, there are at most finitely many repdigit perfect numbers, and the set of all such n...

ON THE EXISTENCE OF e-MULTIPERFECT NUMBERS

The sum of such divisors of N is. denoted by o '(N) , and the number of such divisors by T^\N) . By conventions 1 is an exponential divisor of itself, so that a(l) = 1. The functions j(XN) and o^K^) were introduced in [1] and have been studied in [1] and [2], An integer N is said to be e-perfect whenever o^\N) = 2N9 and g-multiperfect when o^\N) = kN for an integer k > 2. In [1] and [2], severa...

Β-expansions with Deleted Digits for Pisot Numbers Β

An algorithm is given for computing the Hausdorff dimension of the set(s) Λ = Λ(β,D) of real numbers with representations x = ∑∞ n=1 dnβ −n, where each dn ∈ D, a finite set of “digits”, and β > 0 is a Pisot number. The Hausdorff dimension is shown to be log λ/ log β, where λ is the top eigenvalue of a finite 0-1 matrix A, and a simple algorithm for generating A from the data β,D is given.

beta-EXPANSIONS WITH DELETED DIGITS FOR PISOT NUMBERS beta

Distribution of the First Digits of Fibonacci Numbers

In a recent paper [1] , J. L. Brown, Jr., and R. L, Duncan showed that the sequence *{ QnFn I is uniformly distributed modulo 1 (u.d. mod 1), where c/7 denotes the natural logarithm and Fn is the/? Fibonacci number. In this paper we show that some modifications of these ideas have some interesting consequences concerning the distribution of the first digits of the Fibonacci numbers. This also a...