Binomial formulas via divisors of numbers

Inventiones Mathematicae, 1984

Advances in Difference Equations, 2013

It is known that certain convolution sums can be expressed as a combination of divisor functions and Bernoulli formula. One of the main goals in this paper is to establish combinatoric convolution sums for the divisor sumsσ s (n) = d|n (-1) n d-1 d s. Finally, we find a formula of certain combinatoric convolution sums and Bernoulli polynomials. MSC: 11A05; 33E99

CENTRAL EUROPEAN SYMPOSIUM ON THERMOPHYSICS 2019 (CEST)

In "Construction of some new families of Apostol-type numbers and polynomials via Dirichlet character and p-adic q-integrals Turk J Math (2018) 42: 557-577", we defined some new families of numbers and polynomials related to the Dirichlet character of a finite abelian group. In this paper, we investigate some properties and relations for these numbers and polynomials with their series representations and generating functions. By using functional equations of generating functions with their differential equations and series representations of these numbers and polynomials, we derive some formulas including recurrence relations, combinatorial numbers, combinatorial sums and special numbers and polynomials. By using p-adic integral method, we also derive p-adic series representations of these series representations of these numbers and polynomials.

Journal of The London Mathematical Society-second Series, 1974

has shown that the density of binomial coefficients divisible by a prime p is one. Rosch 5], on the basis of numerical evidence, has conjectured that the density of binomial coefficients divisible by any integer d is one. In this note, I shall prove this in the following form. THEOREM 1. For any positive integer d, almost all binomial coefficients are divisible byd. "Almost all" can have many meanings. I shall give proofs for four different meanings, which appear to be the most appropriate. In any case, it is clear that if we have d = Yl Pi e *, then it suffices to show that each p%H divides almost every coefficient, so we may and do restrict our attention to d = p e .

2003

1. The Smarandache, Pseudo-Smarandache, resp. Smarandache-simple functions are defined as ([7J, [6]) S{n) = min{rn EN: nlm!}, Z(n) = min {m.E N: nl m{n~ + 1)} , 5p (n) = min{m EN: p"lm!} for fixed primes p. The duals of Sand Z have been studied e.g. in (2], [5J, [6]: 5.(n) = max{m EN: m!ln}, { m(rn+1) } Z.(n) = max mEN: 2

Colloquium Mathematicum, 2020

Let τ (n) stand for the number of divisors of the positive integer n. We obtain upper bounds for τ (n) in terms of log n and the number of distinct prime factors of n.

2011

Let σ (e) (n) denote the sum of the exponential divisors of n, τ (e) (n) denote the number of the exponential divisors of n, σ (e) ∗ (n) denote the sum of the e-unitary divisors of n and τ (e) ∗ (n) denote the number of the e-unitary divisors of n. The aim of this paper is to present several inequalities about the arithmetic functions which use exponential divisors. Among these inequalities, we have the following: σ (e)(n) τ (e)(n) ≥ γ(n) + τ (e)(n) − 1, for any n ≥ 1, 2 σ(e) ∗ (n) τ (e)∗(n) ≥ γ(n) + τ (e)∗(n) − 1, for any 2 n ≥ 1 and σ(n) + 1 ≥ σ (e) (n) + τ(n), for any n ≥ 1, where τ(n) is the number of the natural divisors of n, σ(n) is the sum of the divisors of n and γ is the ”core ” of n.

Lecture Notes in Mathematics, 1989

Abstract and Applied Analysis, 2014

We study combinatoric convolution sums of certain divisor functions involving even indices. We express them as a linear combination of divisor functions and Euler polynomials and obtain identitiesD2k(n)=(1/4)σ2k+1,0(n;2)-2·42kσ2k+1(n/4) -(1/2)[∑d|n,d≡1 (4){E2k(d)+E2k(d-1)}+22k∑d|n,d≡1 (2)E2k((d+(-1)(d-1)/2)/2)],U2k(p,q)=22k-2[-((p+q)/2)E2k((p+q)/2+1)+((q-p)/2)E2k((q-p)/2)-E2k((p+1)/2)-E2k((q+1)/2)+E2k+1((p+q)/2+1)-E2k+1((q-p)/2)], andF2k(n)=(1/2){σ2k+1†(n)-σ2k†(n)}. As applications of these identities, we give several concrete interpretations in terms of the procedural modelling method.

The purpose of this paper is to present several inequalities about the arithmetic functions σ (e) ,τ (e) ,σ (e)* ,τ (e)* and other well-known arithmetic functions. Among these, we have the following: σ k * (n)·σ l * (n) σ k-l 2 * (n)≤n l-k 4 ·σ k * (n)+n k-l 4 ·σ l * (n) 2·σ k-l 2 (n)≤n l-k 4 ·n k+l 2 +1 2, for any n,k,l∈ℕ * , σ k (e)* (n)·τ (e)* (n) σ k-l 2 (e)* (n)≤n l-k 4 ·σ k (e)* (n)+n k-l 4 ·τ (e)* (n) 2·σ k-l 2 (e)* (n)≤n l-k 4 ·n k+l 2 +1 2, for any n,k,l∈ℕ * , σ k (e) (n)·σ l (e) (n)≤τ (e) (n)·τ k+l (e) (n) for any n,k,l∈ℕ * and σ k+1 (e)* (n) σ k (e)* (n)≥σ (e)* (n) τ (e)* (n)≥τ(n) for any n,k∈ℕ * , where τ(n) is the number of the natural divisors of n and σ(n) is the sum of the divisors of n.