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Vladimir Shevelev

Publications and source records attributed to Vladimir Shevelev.

At least 19 recordsLinked to original sources
arXiv2017

On a Luschny question

Let $E_n(x)$ be Euler polynomial, $ν_2(n)$ be $2-$adic order of $n,$ $\{g(n)\}$ be the characteristic sequence for $\{2^{n}-1\}_{n\geq1}.$ Recently Peter Luschny asked (cf. \cite{5}, sequence A290646): is for $n\geq1$ A290646=A135517? According to A135517, A091090 and a formula there, this question is equivalent to the following one: is, for $n\geq1,$ the denominator of $E_n(x) - E_n(1)$ equal to $2^{ν_2(n+1)-g(n)}$? In this note we answer in the affirmative on this question.

math.NT↗
arXiv2017

Two analogs of Thue-Morse sequence

We introduce and study two analogs of one of the best known sequence in Mathematics : Thue-Morse sequence. The first analog is concerned with the parity of number of runs of 1's in the binary representation of nonnegative integers. The second one is connected with the parity of number of 1's in the representation of nonnegative integers in so-called negabinary (or in base $-2).$ We give for them some recurrent and structure formulas and prove that the second $(0,1)$-sequence is cube-free, while the first one is quint-free. Finally we consider several interesting unsolved problems.

math.NT↗
arXiv2016

On Erdős constant

In 1944, P. Erdős \cite{1} proved that if $n$ is a large highly composite number (HCN) and $n_1$ is the next HCN, then $$n 0$ is a constant. In this paper, using numerical results by D. A. Corneth, we show that most likely $c<1.$

math.NT↗
arXiv2016

Representation of positive integers by the form $x^3+y^3+z^3-3xyz$

We study the number $ν(n)$ of representations of a positive integer $n$ by the form $x^3+y^3+z^3-3xyz$ in the conditions $0\leq x\leq y\leq z; z\geq x+1.$ We proved the following results: (i) for every positive $n,$ except for $n\equiv\pm3 \pmod9,$ $ν(n)>=1;$ (ii) for the exceptional $n,$ $ν(n)=0;$ (iii) for every prime $p\neq3,$ $ν(p)=ν(2p)=1;$ (iv) $\limsup (ν(n))=\infty;$ (v) for every positive $n,$ there exists $k$ such that $ν(k)=n.$

math.NT↗
arXiv2016

Set of all densities of exponentially S-numbers

Let $\mathbf{G}$ be the set of all finite or infinite increasing sequences of positive integers beginning with 1. For a sequence $S=\{s(n)\}, n\geq1,$ from $\mathbf{G},$ a positive number $N$ is called an exponentially $S$-number $(N\in E(S)),$ if all exponents in its prime power factorization are in $S.$ The author \cite{2} proved that, for every sequence $S\in \mathbf{G},$ the sequence of exponentially $S$-numbers has a density $h=h(E(S))\in [\frac{6}{π^2}, 1].$ In this paper we study the set $\{h(E(S)\}$ of all such densities.

math.NT↗
arXiv2016

A fast computation of density of exponentially $S$-numbers

The author \cite{4} proved that, for every set $S$ of positive integers containing 1 (finite or infinite) there exists the density $h=h(E(S))$ of the set $E(S)$ of numbers whose prime factorizations contain exponents only from $S,$ and gave an explicit formula for $h(E(S)).$ In this paper we give an equivalent polynomial formula for $\log h(E(S))$ which allows to get a fast calculation of $h(E(S)).$

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arXiv2016

Exponentially $S$-numbers

Let $\mathbf{S}$ be the set of all finite or infinite increasing sequences of positive integers. For a sequence $S=\{s(n)\}, n\geq1,$ from $\mathbf{S},$ let us call a positive number $N$ an exponentially $S$-number $(N\in E(S)),$ if all exponents in its prime power factorization are in $S.$ Let us accept that $1\in E(S).$ We prove that, for every sequence $S\in \mathbf{S}$ with $s(1)=1,$ the exponentially $S$-numbers have a density $h=h(E(S))$ such that $$\sum_{i\leq x,\enskip i\in E(S)} 1 = h(E(S))x+R(x), where R(x) does not depend on $S$ and $h(E(S))=\prod_{p}(1+\sum_{i\geq2}\frac{u(i)-u(i-1)}{p^i}),$ where $u(n)$ is the characteristic function of $S.$

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arXiv2015

The ménage problem with a known mathematician

We give a solution of the following combinatorial problem: "Let one from $n$ married couples in the ménage problem (see Problem 1) be a couple of a known mathematician $M$ and his wife. After the ladies are seated at every other chair, $M$ (in token of respect) is the first man allowed to choose one of the remaining chairs. To find the number of ways of seating the other men, with no man seated next to his wife, if $M$ chooses the chair that is $d$ seats clockwise from his wife's chair."

math.CO↗
arXiv2015

The Yellowstone Permutation

Define a sequence of positive integers by the rule that a(n) = n for 1 <= n <= 3, and for n >= 4, a(n) is the smallest number not already in the sequence which has a common factor with a(n-2) and is relatively prime to a(n-1). We show that this is a permutation of the positive integers. The remarkable graph of this sequence consists of runs of alternating even and odd numbers, interrupted by small downward spikes followed by large upward spikes, suggesting the eruption of geysers in Yellowstone National Park. On a larger scale the points appear to lie on infinitely many distinct curves. There are several unanswered questions concerning the locations of these spikes and the equations for these curves.

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arXiv2014

Theorems on twin primes-dual case

We prove dual theorems to theorems proved by author in \cite {5}. Beginning with Section 10, we introduce and study so-called "twin numbers of the second kind" and a postulate for them. We give two proofs of the infinity of these numbers and a sufficient condition for truth of the postulate; also we pose several other conjectures. Finally, we consider a conception of axiom of type "AiB".

math.GM↗
arXiv2014

Tangent power sums and their applications

For integer $m, p,$ we study tangent power sum $\sum^m_{k=1}\tan^{2p}\frac{πk}{2m+1}.$ We prove that, for every $m, p,$ it is integer, and, for a fixed p, it is a polynomial in $m$ of degree $2p.$ We give recurrent, asymptotical and explicit formulas for these polynomials and indicate their connections with Newman's digit sums in base $2m.$

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arXiv2014

Combinatorial minors for matrix functions and their applications

As well known, permanent of a square (0,1)-matrix $A$ of order $n$ enumerates the permutations $β$ of $1,2,...,n$ with the incidence matrices $B\leq A.$ To obtain enumerative information on even and odd permutations with condition $B\leq A,$ we should calculate two-fold vector $(a_1,a_2)$ with $a_1+a_2 =per A.$ More general, the introduced $ω$-permanent, where $ω=e^{2πi/m},$ we calculate as $m$-fold vector. For these and other matrix functions we generalize the Laplace theorem of their expansion over elements of the first row, using the defined so-called "combinatorial minors". In particular, in this way, we calculate the cycle index of permutations with condition $B\leq A.$

math.CO↗
arXiv2014

Beyond odious and evil

In a recent post on the Seqfan list the third author proposed a conjecture concerning the summatory function of odious numbers (i.e., of numbers whose sum of binary digits is odd), and its analog for evil numbers (i.e., of numbers whose sum of binary digits is even). We prove these conjectures here. We will also study the sequences of "generalized" odious and evil numbers, and their iterations, giving in particular a characterization of the sequences of usual odious and evil numbers in terms of functional equations satisfied by their compositions.

math.NT↗
arXiv2013

Binary Additive Problems: Recursions for Numbers of Representations

We prove some general recursions for the numbers of representations of positive integers as a sum x+y, x in X, y in Y, where X,Y are increasing sequences. In particular, we obtain recursions for the number of the Goldbach, Lemoine-Levy, Chen and other binary partitions.

math.NT↗
arXiv2012

On intervals (kn,(k+1)n) containing a prime for all n>1

We study values of k for which the interval (kn,(k+1)n) contains a prime for every n>1. We prove that the list of such integers k includes k=1,2,3,5,9,14, and no others, at least for k<=50,000,000. For every known k of this list, we give a good upper estimate of the smallest N_k(m), such that, if n>=N_k(m), then the interval (kn,(k+1)n) contains at least m primes.

math.NT↗