Multiple Perfect Numbers, Mersenne Primes, and Effective Computability.
On the distribution of amicable numbers. II.
On the distribution of amicable numbers.
On the composition of the arithmetic functions σ and φ
Odd perfect numbers are divisible by at least seven distinct primes
On composite n for which φ(n) | n -1
On the distribution of the values of Euler's function
On the congruences and
On primitive divisors of Mersenne numbers
Vyprávění o dvou sítech
On the largest prime factors of n and n+1. (Short Communication).
On the largest prime factors of n and n+1.
On the number of distinct values of Euler's φ-function
On the range of Carmichael's universal-exponent function
Let λ denote Carmichael’s function, so λ(n) is the universal exponent for the multiplicative group modulo n. It is closely related to Euler’s φ-function, but we show here that the image of λ is much denser than the image of φ. In particular the number of λ-values to x exceeds for all large x, while for φ it is equal to , an old result of Erdős. We also improve on an earlier result of the first author and Friedlander giving an upper bound for the distribution of λ-values.
The iterated Carmichael λ-function and the number of cycles of the power generator
On the periods of the linear congruential and power generators
Acknowledgment of priority: "On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions ϕ and σ" (Colloq. Math. 92 (2002), 111-130)
On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions ϕ and σ
Let σ(n) denote the sum of positive divisors of the integer n, and let ϕ denote Euler's function, that is, ϕ(n) is the number of integers in the interval [1,n] that are relatively prime to n. It has been conjectured by Mąkowski and Schinzel that σ(ϕ(n))/n ≥ 1/2 for all n. We show that σ(ϕ(n))/n → ∞ on a set of numbers n of asymptotic density 1. In addition, we study the average order of σ(ϕ(n))/n as well as its range. We use similar methods to prove a conjecture of Erdős that ϕ(n-ϕ(n)) < ϕ(n)...
On the distribution of some integers related to perfect and amicable numbers
The range of the sum-of-proper-divisors function
Answering a question of Erdős, we show that a positive proportion of even numbers are in the form s(n), where s(n) = σ(n) - n, the sum of proper divisors of n.
Page 1 Next