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A set $𝒮 = {x_{1}, ..., x_{n}}$ of $n$ distinct positive integers is said to be gcd-closed if $(x_{i}, x_{j}) \in 𝒮$ for all $1 \leq i, j \leq n$ . Shaofang Hong conjectured in 2002 that for a given positive integer $t$ there is a positive integer $k (t)$ depending only on $t$ , such that if $n \leq k (t)$ , then the power LCM matrix $({[x_{i}, x_{j}]}^{t})$ defined on any gcd-closed set $𝒮 = {x_{1}, ..., x_{n}}$ is nonsingular, but for $n \geq k (t) + 1$ , there exists a gcd-closed set $𝒮 = {x_{1}, ..., x_{n}}$ such that the power LCM matrix $({[x_{i}, x_{j}]}^{t})$ on $𝒮$ is singular. In 1996, Hong proved $k (1) = 7$ and noted $k (t) \geq 7$ for all $t \geq 2$ . This paper develops Hong’s method and provides a new idea to calculate...

On integers n with $J_{t} (n) < J_{t} (m)$ for $m > n$ .

Chidambaraswamy, J., Krishnaiah, P.V. (1989)

International Journal of Mathematics and Mathematical Sciences

On integers not of the form n - φ (n)

J. Browkin, A. Schinzel (1995)

Colloquium Mathematicae

W. Sierpiński asked in 1959 (see [4], pp. 200-201, cf. [2]) whether there exist infinitely many positive integers not of the form n - φ(n), where φ is the Euler function. We answer this question in the affirmative by proving Theorem. None of the numbers $2^{k} \cdot 509203$ (k = 1, 2,...) is of the form n - φ(n).

On Jacobsthal's g(n)-Function.

Harlan Stevens (1977)

Mathematische Annalen

On $k$ -imperfect numbers.

Zhou, Weiyi, Zhu, Long (2009)

Integers

On multiplicatively e-perfect numbers.

Sándor, József (2004)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

On multiplicatively perfect numbers.

Sándor, J. (2001)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

On $n ∣ ϕ (n) d (n) + 2$ and $n ∣ ϕ (n) σ (n) + 1$ .

Chen, Yong-Gao, Fang, Jin-Hui (2008)

Integers

On near-perfect and deficient-perfect numbers

Min Tang, Xiao-Zhi Ren, Meng Li (2013)

Colloquium Mathematicae

For a positive integer n, let σ(n) denote the sum of the positive divisors of n. Let d be a proper divisor of n. We call n a near-perfect number if σ(n) = 2n + d, and a deficient-perfect number if σ(n) = 2n - d. We show that there is no odd near-perfect number with three distinct prime divisors and determine all deficient-perfect numbers with at most two distinct prime factors.

On near-perfect numbers

Min Tang, Xiaoyan Ma, Min Feng (2016)

Colloquium Mathematicae

For a positive integer n, let σ(n) denote the sum of the positive divisors of n. We call n a near-perfect number if σ(n) = 2n + d where d is a proper divisor of n. We show that the only odd near-perfect number with four distinct prime divisors is 3⁴·7²·11²·19².

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On Dirichlet characters of polynomials

On distribution of values of σ(n) in residue classes

On $f$ -thin sets

On factorization of integers with restrictions on the exponents.

On generalized Landau identities.

On Hong’s conjecture for power LCM matrices

On integers n with $J_{t} (n) < J_{t} (m)$ for $m > n$ .

On integers not of the form n - φ (n)

On Jacobsthal's g(n)-Function.

On $k$ -imperfect numbers.

On multiplicatively e-perfect numbers.

On multiplicatively perfect numbers.

On $n ∣ ϕ (n) d (n) + 2$ and $n ∣ ϕ (n) σ (n) + 1$ .

On near-perfect and deficient-perfect numbers

On near-perfect numbers

On partition functions and divisor sums.

On perfect numbers connected with the composition of arithmetic functions.

On perfect totient numbers.

On periodic mod p sequences and G-functions (On a conjecture of Ruzsa).

On recurrent mod p sequences.

Currently displaying 241 – 260 of 487