Abstract
For each integer n , denote by p p n the largest prime ≤ n and by P p n the smallest prime ≥ n , called as the Smarandache inferior prime part and superior prime part of n , respectively. Define I n ≔ 1 / n ∑ m ≤ n p p m and S n ≔ 1 / n ∑ m ≤ n P p m . In this short note, we proved some estimates on I n − S n and I n / S n , answering a question proposed by Kashihara and improving a result of Yan.
Highlights
Contains infinitely many primes. is result is known as Dirichlet’s theorem
Is called the associated Betty sequence. It follows from a classical result of Ivan Vinogradov that the number πB(α,β) ∼ (1/α)π(x)
Smarandache published a book named Only problems, not solutions! In this book, he presented 105 unsolved arithmetical problems and conjectures about special sequences and functions, which can help us to analyze the properties of primes and the factors of integers
Summary
Contains infinitely many primes. is result is known as Dirichlet’s theorem. Take another example; for any positive real numbers α and β, the set. It follows from a classical result of Ivan Vinogradov that the number πB(α,β) ∼ (1/α)π(x). Smarandache published a book named Only problems, not solutions! He presented 105 unsolved arithmetical problems and conjectures about special sequences and functions, which can help us to analyze the properties of primes and the factors of integers.
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