In this paper, we prove that, if $N$ is a positive odd number with $r$ distinct prime factors such that $N\mid \sigma (N)$, then $N\lt {2}^{{4}^{r} - {2}^{r} } $ and $N{\mathop{\prod }\nolimits}_{p\mid N} p\lt {2}^{{4}^{r} } $, where $\sigma (N)$ is the sum of all positive divisors of $N$. In particular, these bounds hold if $N$ is an odd perfect number.

In 2007, Andrews and Paule introduced a new class of combinatorial objects called broken $k$-diamond partitions. Recently, Shishuo Fu generalised the notion of broken $k$-diamond partitions to combinatorial objects which he termed $k$ dots bracelet partitions. Fu denoted the number of $k$ dots bracelet partitions of $n$ by ${\mathfrak{B}}_{k} (n)$ and proved several congruences modulo primes and modulo powers of 2. More recently, Radu and Sellers extended the set of congruences proven by Fu by proving three congruences modulo squares of primes for ${\mathfrak{B}}_{5} (n)$, ${\mathfrak{B}}_{7} (n)$ and ${\mathfrak{B}}_{11} (n)$. In this note, we prove some congruences modulo powers of 2 for ${\mathfrak{B}}_{5} (n)$. For example, we find that for all integers $n\geq 0$, ${\mathfrak{B}}_{5} (16n+ 7)\equiv 0\hspace{0.167em} ({\rm mod} \hspace{0.334em} {2}^{5} )$.

All groups considered in this paper are finite. A subgroup $H$ of a group $G$ is called a primitive subgroup if it is a proper subgroup in the intersection of all subgroups of $G$ containing $H$ as a proper subgroup. He et al. [‘A note on primitive subgroups of finite groups’, Commun. Korean Math. Soc. 28(1) (2013), 55–62] proved that every primitive subgroup of $G$ has index a power of a prime if and only if $G/ \Phi (G)$ is a solvable PST-group. Let $\mathfrak{X}$ denote the class of groups $G$ all of whose primitive subgroups have prime power index. It is established here that a group $G$ is a solvable PST-group if and only if every subgroup of $G$ is an $\mathfrak{X}$-group.

The long-standing open problem of finding an upper bound for the Wiener index of a graph in terms of its order and diameter is addressed. Sharp upper bounds are presented for the Wiener index, and the related degree distance and Gutman index, for trees of order $n$ and diameter at most $6$.

Let $\Omega $ be the set of unit vectors and $w$ be a radial weight on the plane. We consider the weighted directional maximal operator defined by

$$\begin{eqnarray*}{M}_{\Omega , w} f(x): = \sup _{x\in R\in \mathcal{B} _{\Omega }}\frac{1}{w(R)} \int \nolimits \nolimits_{R} \vert f(y)\vert w(y)\hspace{0.167em} dy,\end{eqnarray*}$$

${ \mathcal{B} }_{\Omega } $

$\Omega $

$w(R)$

$\int \nolimits \nolimits_{R} w$

$$\begin{eqnarray*}\Vert {M}_{\Omega , w} f\mathop{\Vert }\nolimits_{{L}^{2} (w)} \leq C\log N\Vert f\mathop{\Vert }\nolimits_{{L}^{2} (w)} ,\end{eqnarray*}$$

$w(x)= \vert x\hspace{-1.2pt}\mathop{\vert }\nolimits ^{a} $

$a\gt 0$

$\Omega $

$N$

In the 2006 edition of the Kourovka Notebook, Berkovich poses the following problem (Problem 16.13): Let $p$ be a prime and $P$ be a finite $p$-group. Can $P$ have every maximal subgroup special? We show that the structure of such groups is very restricted, but for all primes there are groups of arbitrarily large size with this property.

In this paper, we construct several new permutation polynomials over finite fields. First, using the linearised polynomials, we construct the permutation polynomial of the form ${ \mathop{\sum }\nolimits}_{i= 1}^{k} ({L}_{i} (x)+ {\gamma }_{i} ){h}_{i} (B(x))$ over ${\mathbf{F} }_{{q}^{m} } $, where ${L}_{i} (x)$ and $B(x)$ are linearised polynomials. This extends a theorem of Coulter, Henderson and Matthews. Consequently, we generalise a result of Marcos by constructing permutation polynomials of the forms $xh({\lambda }_{j} (x))$ and $xh({\mu }_{j} (x))$, where ${\lambda }_{j} (x)$ is the $j$th elementary symmetric polynomial of $x, {x}^{q} , \ldots , {x}^{{q}^{m- 1} } $ and ${\mu }_{j} (x)= {\mathrm{Tr} }_{{\mathbf{F} }_{{q}^{m} } / {\mathbf{F} }_{q} } ({x}^{j} )$. This answers an open problem raised by Zieve in 2010. Finally, by using the linear translator, we construct the permutation polynomial of the form ${L}_{1} (x)+ {L}_{2} (\gamma )h(f(x))$ over ${\mathbf{F} }_{{q}^{m} } $, which extends a result of Kyureghyan.

An $r$-ary necklace (bracelet) of length $n$ is an equivalence class of $r$-colourings of vertices of a regular $n$-gon, taking all rotations (rotations and reflections) as equivalent. A necklace (bracelet) is symmetric if a corresponding colouring is invariant under some reflection. We show that the number of symmetric $r$-ary necklaces (bracelets) of length $n$ is $\frac{1}{2} (r+ 1){r}^{n/ 2} $ if $n$ is even, and ${r}^{(n+ 1)/ 2} $ if $n$ is odd.

Let $\sigma (n)= {\mathop{\sum }\nolimits}_{d\mid n} d$ be the usual sum-of-divisors function. In 1933, Davenport showed that $n/ \sigma (n)$ possesses a continuous distribution function. In other words, the limit $D(u): = \lim _{x\rightarrow \infty }(1/ x){\mathop{\sum }\nolimits}_{n\leq x, n/ \sigma (n)\leq u} 1$ exists for all $u\in [0, 1] $ and varies continuously with $u$. We study the behaviour of the sums ${\mathop{\sum }\nolimits}_{n\leq x, n/ \sigma (n)\leq u} f(n)$ for certain complex-valued multiplicative functions $f$. Our results cover many of the more frequently encountered functions, including $\varphi (n)$, $\tau (n)$ and $\mu (n)$. They also apply to the representation function for sums of two squares, yielding the following analogue of Davenport’s result: for all $u\in [0, 1] $, the limit

$$\begin{eqnarray*}\tilde {D} (u): = \lim _{R\rightarrow \infty }\frac{1}{\pi R} \# \biggl\{ (x, y)\in { \mathbb{Z} }^{2} : 0\lt {x}^{2} + {y}^{2} \leq R\text{ and } \frac{{x}^{2} + {y}^{2} }{\sigma ({x}^{2} + {y}^{2} )} \leq u\biggr\}\end{eqnarray*}$$

$\tilde {D} (u)$

$[0, 1] $

We consider the incidence semirings of graphs and prove that every incidence semiring has convenient visible bases for its right ideals and for its left ideals, and that these visible bases can be used to determine the weights of all right ideals that have maximum weight and all left ideals that have maximum weight.

For $n\in \mathbb{Z} $ and $A\subseteq \mathbb{Z} $, let ${r}_{A} (n)= \# \{ ({a}_{1} , {a}_{2} )\in {A}^{2} : n= {a}_{1} + {a}_{2} , {a}_{1} \leq {a}_{2} \} $ and ${\delta }_{A} (n)= \# \{ ({a}_{1} , {a}_{2} )\in {A}^{2} : n= {a}_{1} - {a}_{2} \} $. We call $A$ a unique representation bi-basis if ${r}_{A} (n)= 1$ for all $n\in \mathbb{Z} $ and ${\delta }_{A} (n)= 1$ for all $n\in \mathbb{Z} \setminus \{ 0\} $. In this paper, we construct a unique representation bi-basis of $ \mathbb{Z} $ whose growth is logarithmic.

Connections between annihilators and ideals in Frobenius and symmetric algebras are used to provide a new proof of a result of Nakayama on quotient algebras, and an application is given to central symmetric algebras.

Blecher [‘Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal’, Util. Math. 88 (2012), 223–235] defined the combinatorial objects known as 1-shell totally symmetric plane partitions of weight $n$. He also proved that the generating function for $f(n), $ the number of 1-shell totally symmetric plane partitions of weight $n$, is given by

$$\begin{eqnarray*}\displaystyle \sum _{n\geq 0}f(n){q}^{n} = 1+ \sum _{n\geq 1}{q}^{3n- 2} \prod _{i= 0}^{n- 2} (1+ {q}^{6i+ 3} ).\end{eqnarray*}$$

$f(n)$

If $X$ is a subgroup of a group $G$, the cardinal number $\min \{ \vert X: X_{G}\vert , \vert {X}^{G} : X\vert \} $ is called the normal oscillation of $X$ in $G$. It is proved that if all subgroups of a locally finite group $G$ have finite normal oscillation, then $G$ contains a nilpotent subgroup of finite index.

Let $\alpha : 1, 1, \sqrt{x} , \mathop{( \sqrt{u} , \sqrt{v} , \sqrt{w} )}\nolimits ^{\wedge } $ be a backward 3-step extension of a recursively generated weighted sequence of positive real numbers with $1\leq x\leq u\leq v\leq w$ and let ${W}_{\alpha } $ be the associated weighted shift with weight sequence $\alpha $. The set of positive real numbers $x$ such that ${W}_{\alpha } $ is quadratically hyponormal for some $u, v$ and $w$ is described, solving an open problem due to Curto and Jung [‘Quadratically hyponormal weighted shifts with two equal weights’, Integr. Equ. Oper. Theory 37 (2000), 208–231].

Answering a problem posed by John Michael Rassias, we study the functional inequality

$$\begin{eqnarray*}f(x+ y+ xy)\leq f(x)+ f(y)+ f(xy),\end{eqnarray*}$$

$f$

Suppose that $k_0\geq 3.5\times 10^6$ and $\mathcal{H}=\{h_1,\ldots,h_{k_0}\}$ is admissible. Then, for any $m\geq 1$, the set $\{m(h_j-h_i):\, h_i<h_j\}$ contains at least one Polignac number.

If $T$ is any finite higher commutator in an associative ring $R$, for example, $T= [[R, R] , [R, R] ] $, and if $T$ has minimal cardinality so that the ideal generated by $T$ is infinite, then $T$ is in the centre of $R$ and ${T}^{2} = 0$. Also, if $T$ is any finite, higher commutator containing no nonzero nilpotent element then $T$ generates a finite ideal.

We mainly prove that, assuming $\mathfrak{b}= \mathfrak{c}$, every regular star-compact space with a strong rank 1-diagonal is metrisable.

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be $E$-supplemented in $G$ if there is a subgroup $T$ of $G$ such that $G= HT$ and $H\cap T\leq {H}_{eG} $, where ${H}_{eG} $ denotes the subgroup of $H$ generated by all those subgroups of $H$ which are $S$-quasinormally embedded in $G$. In this paper, some new characterisations of $p$-supersolubility of finite groups are given under the assumption that some primary subgroups are $E$-supplemented.