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Register a new userThe Repeated Divisor Function and Possible Correlation with Highly Composite Numbers
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Let n be a non-null positive integer and $d(n)$ is the number of positive divisors of n, called the divisor function. Of course, $d(n) leq n$. $d(n) = 1$ if and only if $n = 1$. For $n > 2$ we have $d(n) geq 2$ and in this paper we try to find the smallest $k$ such that $d(d(...d(n)...)) = 2$ where the divisor function is applied $k$ times. At the end of the paper we make a conjecture based on some observations.
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Let $alpha=0.a_1a_2a_3ldots$ be an irrational number in base $b>1$, where $0leq a_i<b$. The number $alpha in (0,1)$ is a $textit{normal number}$ if every block $(a_{n+1}a_{n+2}ldots a_{n+k})$ of $k$ digits occurs with probability $1/b^k$. A conditional proof of the normality of the real number $pi$ in base $10$ is presented in this note.
The proofs that the real numbers are denumerable will be shown, i.e., that there exists one-to-one correspondence between the natural numbers $N$ and the real numbers $Re$. The general element of the sequence that contains all real numbers will be explicitly specified, and the first few elements of the sequence will be written. Remarks on the Cantors nondenumerability proofs of 1873 and 1891 that the real numbers are noncountable will be given.
Let $ xgeq 1 $ be a large number, let $ [x]=x-{x} $ be the largest integer function, and let $ varphi(n)$ be the Euler totient function. The result $ sum_{nleq x}varphi([x/n])=(6/pi^2)xlog x+Oleft ( x(log x)^{2/3}(loglog x)^{1/3}right ) $ was proved very recently. This note presents a short elementary proof, and sharpen the error term to $ sum_{nleq x}varphi([x/n])=(6/pi^2)xlog x+O(x) $. In addition, the first proofs of the asymptotics formulas for the finite sums $ sum_{nleq x}psi([x/n])=(15/pi^2)xlog x+O(xlog log x) $, and $ sum_{nleq x}sigma([x/n])=(pi^2/6)xlog x+O(x log log x) $ are also evaluated here.
Some properties of the optimal representation of numbers are investigated. This representation, which is to the base-e, is examined for coding of integers. An approximate representation without fractions that we call WF is introduced and compared with base-2 and base-3 representations, which are next to base-e in efficiency. Since trees are analogous to number representation, we explore the relevance of the statistical optimality of the base-e system for the understanding of complex system behavior and of social networks. We show that this provides a new theoretical explanation for the nature of the power law exhibited by many open complex systems. In specific, we show that the power law distribution most often proposed for such systems has a form that is similar to that derived from the optimal base-e representation.
Let $ xgeq 1 $ be a large number, let $ [x]=x-{x} $ be the largest integer function, and let $ sigma(n)$ be the sum of divisors function. This note presents the first proof of the asymptotic formula for the average order $ sum_{pleq x}sigma([x/p])=c_0xlog log x+O(x) $ over the primes, where $c_0>0$ is a constant. More generally, $ sum_{pleq x}sigma([x/(p+a)])=c_0xlog log x+O(x) $ for any fixed integer $a$.
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