Transition Metal Free Chemoselective Reduction of α,β‐Unsaturated Ketones to Saturated Ketones Using Tosyl Hydrazide as a Hydrogen Donor

An efficient, inexpensive and simple method for the reduction of various α,β‐unsaturated ketones to corresponding saturated ketones using tosyl hydrazide as a hydrogen donor in DMF using calcium oxide powder has been reported. A variety of enones underwent reduction without forming undesirable side products. High chemoselectivity, broad functional group tolerance and good yields are the noteworthy features of this protocol.     

Congruences for 7 and 49-regular partitions modulo powers of 7

Let \(b_{k}(n)\) denote the number of k-regular partitions of n. In this paper, we prove Ramanujan-type congruences modulo powers of 7 for \(b_{7}(n)\) and \(b_{49}(n)\). For example, for all \(j\ge 1\) and \(n\ge 0\), we prove that

$$\begin{aligned} b_{7}\Bigg (7^{2j-1}n+\frac{3\cdot 7^{2j-1}-1}{4}\Bigg )\equiv 0\pmod {7^{j}} \end{aligned}$$

and

$$\begin{aligned} b_{49}\Big (7^{j}n+7^{j}-2\Big )\equiv 0\pmod {7^{j}}. \end{aligned}$$

     

On overpartition pairs into odd parts modulo powers of 2

In 2012, Lin (Electron. J. Combin. 19(2) (2012) #P17) investigated the 2 and 3-divisibility properties for ${\overline{pp}}_o\left(n\right)$, the number of overpartition pairs into odd parts. Using modular forms, he proved that for a fixed positive integer $k$, ${\overline{pp}}_o\left(n\right)$ is almost always divisible by $2^k$. In this paper, we prove several congruences for ${\overline{pp}}_o\left(n\right)$ modulo higher powers of 2 in an elementary way.     

On 3-Regular Tripartitions

Abstract In this article,we investigate the arithmetic behavior of the function D_3(n)which counts the number of 3-regular tripartitions of n.For example,we show that forα≥1 and n≥0,■and ■     

Some properties of k-tuple t-core partitions

The Ramanujan Journal - An overpartition is a partition in which the first occurrence of a number may be overlined. For an overpartition $$\lambda $$ , let $$\ell (\lambda )$$ denote the largest...     

Modular relations for the Rogers–Ramanujan functions with applications to partitions

The Ramanujan Journal - In this paper, we obtain several new modular relations for the Rogers–Ramanujan functions. Furthermore, we give partition theoretic interpretations for some of our...