Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAn identity involving number of representations of $n$ as a sum of $r$ triangular numbers
90
0
0.0
(
0
)
Authors
Sumit Kumar Jha
Ask ChatGPT about the research
No Arabic abstract
Let $sum_{d|n}$ denote sum over divisors of a positive integer $n$, and $t_{r}(n)$ denote the number of representations of $n$ as a sum of $r$ triangular numbers. Then we prove that $$ sum_{d|n}frac{1+2,(-1)^{d}}{d}=sum_{r=1}^{n}frac{(-1)^{r}}{r}, binom{n}{r}, t_{r}(n) $$ using a result of Ono, Robbins and Wahl.
rate research
Read More
In this paper we give a definition of cyclic orthonormal generators (cogs) in R^N. We give a general canonical form for their expression. Further, we give an explicit formula for computing the canonical form of any given cog.
We present a common ground for infinite sums, unordered sums, Riemann integrals, arc length and some generalized means. It is based on extending functions on finite sets using Hausdorff metric in a natural way.
We propose a new class of algebraic structure named as emph{$(m,n)$-semihyperring} which is a generalization of usual emph{semihyperring}. We define the basic properties of $(m,n)$-semihyperring like identity elements, weak distributive $(m,n)$-semihyperring, zero sum free, additively idempotent, hyperideals, homomorphism, inclusion homomorphism, congruence relation, quotient $(m,n)$-semihyperring etc. We propose some lemmas and theorems on homomorphism, congruence relation, quotient $(m,n)$-semihyperring, etc and prove these theorems. We further extend it to introduce the relationship between fuzzy sets and $(m,n)$-semihyperrings and propose fuzzy hyperideals and homomorphism theorems on fuzzy $(m,n)$-semihyperrings and the relationship between fuzzy $(m,n)$-semihyperrings and the usual $(m,n)$-semihyperrings.
We present in this work a heuristic expression for the density of prime numbers. Our expression leads to results which possesses approximately the same precision of the Riemanns function in the domain that goes from 2 to 1010 at least. Instead of using a constant as was done by Legendre and others in the formula of Gauss, we try to adjust the data through a function. This function has the remarkable property: its points of discontinuity are the prime numbers.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
