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Sai Teja Somu

Publications and source records attributed to Sai Teja Somu.

13 recordsLinked to original sources
arXiv2024

On Sums of Practical Numbers and Polygonal Numbers

Practical numbers are positive integers $n$ such that every positive integer less than or equal to $n$ can be written as a sum of distinct positive divisors of $n$. In this paper, we show that all positive integers can be written as a sum of a practical number and a triangular number, resolving a conjecture by Sun. We also show that all sufficiently large natural numbers can be written as a sum of a practical number and two $s$-gonal numbers.

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arXiv2023

Some Results on Zumkeller Numbers

A positive integer $n$ is said to be a Zumkeller number or an integer-perfect number if the set of its positive divisors can be partitioned into two subsets of equal sums. In this paper, we prove several results regarding Zumkeller numbers. For any positive integer $m$, we prove that there are infinitely many positive integers $n$ for which $n+1,\cdots, n+m$ are all Zumkeller numbers. Additionally, we show that every positive integer greater than $94185$ can be expressed as a sum of two Zumkeller numbers and that all sufficiently large integers can be written as a sum of a Zumkeller number and a practical number. We also show that there are infinitely many positive integers that cannot be expressed as a sum of a Zumkeller number and a square or a prime.

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arXiv2022

On Some Results on Practical Numbers

A positive integer $n$ is said to be a practical number if every integer in $[1,n]$ can be represented as the sum of distinct divisors of $n$. In this article, we consider practical numbers of a given polynomial form. We give a necessary and sufficient condition on coefficients $a$ and $b$ for there to be infinitely many practical numbers of the form $an+b$. We also give a necessary and sufficient for a quadratic polynomial to contain infinitely many practical numbers, using which we solve first part of a conjecture mentioned in [9]. In the final section, we prove that every number of $8k+1$ form can be expressed as a sum of a practical number and a square, and for every $j\in \{0,\ldots,7\}\setminus \{1\}$ there are infinitely many natural numbers of $8k+j$ form which cannot be written as sum of a square and a practical number.

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arXiv2022

On Power Values of Sum of Divisors function in Arithmetic Progressions

Let $a\geq 1, b\geq 0$ and $k\geq 2$ be any given integers. It has been proven that there exist infinitely many natural numbers $m$ such that sum of divisors of $m$ is a perfect $k$th power. We try to generalize this result when the values of $m$ belong to any given infinite arithmetic progression $an+b$. We prove if $a$ is relatively prime to $b$ and order of $b$ modulo $a$ is relatively prime to $k$ then there exist infinitely many natural numbers $n$ such that sum of divisors of $an+b$ is a perfect $k$th power. We also prove that, in general, either sum of divisors of $an+b$ is not a perfect $k$th power for any natural number $n$ or sum of divisors of $an+b$ is a perfect $k$th power for infinitely many natural numbers $n$.

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arXiv2021

On a Generalization of Tupper's Formula for $m$ Colours and $n$ Dimensions

Tupper's formula $\frac{1}{2}<\bigg\lfloor \bmod \bigg(\lfloor \frac{y}{17}\rfloor 2^{-17\lfloor x \rfloor -\bmod (\lfloor y \rfloor,17)},2\bigg)\bigg\rfloor$ has an interesting property that for any monochrome image that can be represented by pixels in a two dimensional array of dimensions $106\times 17$, there exists a natural number $k$ such that the graph of the equation in the range $0\leq x <106$ and $k\leq y<k+17$, is that image. In this paper, we give a generalization for $m$ colours and $n$ dimensions. We give $m$ formulae consisting of $n$ free variables, with the property that, for any $n$ dimensional object of $m$ colours $C_1,\cdots, C_m$, that can be represented by hypervoxels(multidimensional analogue of pixel) in a $n$ dimensional array of dimensions $A_1\times \cdots \times A_n$, there exists a natural number $k$ such that, when the first formula is graphed using colour $C_1$, second formula is graphed using colour $C_2$,$\cdots$, $m$th formula is graphed using colour $C_m$ in the range $0\leq x_1<A_1$,$0\leq x_2<A_2,\cdots, 0\leq x_{n-1}<A_{n-1},k\leq x_n <k+A_n$, the union of all graphs is that $n$-dimensional object.

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arXiv2019

Additive Complements for a given Asymptotic Density

{The first version of this text was written and submitted to a journal on April, 12, 2018. This second version was submitted on April, 9, 2019.} We investigate the existence of subsets $A$ and $B$ of $\mathbb{N}:=\{0,1,2,\dots\}$ such that the sumset $A+B:=\{a+b~;a\in A,b\in B\}$ has given asymptotic density. We solve the particular case in which $B$ is a given finite subset of $\mathbb{N}$ and also the case when $B=A$ ; in the later case, we generalize our result to $kA:=\{x_1+\cdots+x_k: x_i\in A, i=1,\dots,k\}$ for an integer $k\geq2.$

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arXiv2017

On $rth$ coefficient of divisors of $x^n-1$

Let $r,n$ be two natural numbers and let $H(r,n)$ denote the maximal absolute value of $r$th coefficient of divisors of $x^n-1$. In this paper, we show that $\sum_{n\leq x}H(r,n)$ is asymptotically equal to $c(r)x(\log x)^{2^r-1}$ for some constant $c(r)>0$. Furthermore, we give an explicit expression of $c(r)$ in terms of $r$.

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arXiv2015

On the coefficients of divisors of $x^n-1$

Let $a(r,n)$ be $r$th coefficient of $n$th cyclotomic polynomial. Suzuki proved that $\{a(r,n)|r\geq 1,n\geq 1\}=\mathbb{Z}$. If $m$ and $n$ are two natural numbers we prove an analogue of Suzuki's theorem for divisors of $x^n-1$ with exactly $m$ irreducible factors. We prove that for every finite sequence of integers $n_1,\ldots,n_r$ there exists a divisor $f(x)=\sum_{i=0}^{deg(f)}c_ix^i$ of $x^n-1$ for some $n\in \mathbb{N}$ such that $c_i=n_i$ for $1\leq i \leq r$. Let $H(r,n)$ denote the maximum absolute value of $r$th coefficient of divisors of $x^n-1$. In the last section of the paper we give tight bounds for $H(r,n)$.

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arXiv2015

On the distribution of numbers related to the divisors of $x^n-1$

Let $n_1,\cdots,n_r$ be any finite sequence of integers and let $S$ be the set of all natural numbers $n$ for which there exists a divisor $d(x)=1+\sum_{i=1}^{deg(d)}c_ix^i$ of $x^n-1$ such that $c_i=n_i$ for $1\leq i \leq r$. In this paper we show that the set $S$ has a natural density. Furthermore, we find the value of the natural density of $S$.

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arXiv2015

On Integer sequences in Product sets

Let $B$ be a finite set of natural numbers or complex numbers. Product set corresponding to $B$ is defined by $B.B:=\{ab:a,b\in B\}$. In this paper we give an upper bound for longest length of consecutive terms of a polynomial sequence present in a product set accurate up to a positive constant. We give a sharp bound on the maximum number of Fibonacci numbers present in a product set when $B$ is a set of natural numbers and a bound which is accurate up to a positive constant when $B$ is a set of complex numbers.

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arXiv2015

On Newman's phenomenon in higher bases

A well known result of Newman says that upto a limit, multiples of $3$ with even number of 1's in binary representation always exceed multiples of $3$ with odd number of 1's. The phenomenon of preponderance of even number of 1's is now known as Newman's phenomenon. We show that this phenomenon exists for higher bases. Let $b$ be a positive integer($\geq 2$). Let $A_{b}$ be the set of all natural numbers which contain only 0's and 1's in b-ary expansion and $S^{(b)}_{q,i}(n)$ be the difference between the corresponding number of $k_e 0$ for sufficiently large $n$. We show that there is a stronger Newman's phenomenon in $A_b$ in the following sense. If $b>2$ and $n=\sum_{i=0}^{k-1}b_i2^i$ with $b_i\in \{0,1\}$, let $b(n)=\sum_{i=0}^{k-1}b_ib^i$ then $\lim_{n\rightarrow \infty} \frac{S^{(2)}_{3,0}(n)}{S^{(b)}_{b+1,0}(b(n))}=0$. That is, for the same number of terms there is stronger preponderance in $A_b$ than in $A_2=\mathbb{N}$. In the last section we show that number of primes $p\leq x$ for which $S_{p,0}^{(b)}(n)>0$ for sufficiently large $n$ is $o\left(\frac{x}{\log x}\right)$.

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arXiv2015

On unavoidable obstructions in Gaussian walks

In this paper we investigate a problem about certain walks in the ring of Gaussian integers. Let $n,d$ be two natural numbers. Does there exist a sequence of Gaussian integers $z_j$ such that $|z_{j+1}-z_j|=1$ and a pair of indices $r$ and $s$, such that $z_{r}-z_{s}=n$ and for all indices $t$ and $u$, $z_{t}-z_{u}\neq d$? If there exists such a sequence we call $n$ to be $d$ avoidable. Let $A_n$ be the set of all $d\in \mathbb{N}$ such that $n$ is not $d$ avoidable. Recently, Ledoan and Zaharescu proved that $\{d \in \mathbb{N} : d|n\}\subset A_n$. We extend this result by giving a necessary and sufficient condition for $d\in A_n$ which answers a question posed by Ledoan and Zaharescu. We also find a precise formula for the cardinality of $A_n$ and answer three other questions raised in the same paper.

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