On consecutive integer pairs with the same sum of distinct prime divisors.
On Dirichlet characters of polynomials
On distribution of values of σ(n) in residue classes
On -thin sets
On factorization of integers with restrictions on the exponents.
On generalized Landau identities.
On Hong’s conjecture for power LCM matrices
A set of distinct positive integers is said to be gcd-closed if for all . Shaofang Hong conjectured in 2002 that for a given positive integer there is a positive integer depending only on , such that if , then the power LCM matrix defined on any gcd-closed set is nonsingular, but for , there exists a gcd-closed set such that the power LCM matrix on is singular. In 1996, Hong proved and noted for all . This paper develops Hong’s method and provides a new idea to calculate...
On integers n with for .
On integers not of the form n - φ (n)
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 (k = 1, 2,...) is of the form n - φ(n).
On Jacobsthal's g(n)-Function.
On -imperfect numbers.
On multiplicatively e-perfect numbers.
On multiplicatively perfect numbers.
On and .
On near-perfect and deficient-perfect numbers
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
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.
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).