When the Sum of Aliquots Divides the Totient

نویسندگان

  • WILLIAM D. BANKS
  • FLORIAN LUCA
چکیده

Let φ(·) be the Euler function and let σ(·) be the sum-of-divisors function. In this note, we bound the number of positive integers n x with the property that s(n) = σ(n)− n divides φ(n).

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

A remark on Giuga’s conjecture and Lehmer’s totient problem∗

Giuga has conjectured that if the sum of the (n− 1)-st powers of the residues modulo n is −1 (mod n), then n is 1 or prime. It is known that any counterexample is a Carmichael number. Lehmer has asked if φ(n) divides n−1, with φ being Euler’s function, must it be true that n is 1 or prime. No examples are known, but a composite number with this property must be a Carmichael number. We show that...

متن کامل

Common values of the arithmetic functions φ and σ Kevin

We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sum-of-divisors function. This proves a fifty-year-old conjecture of Erdős. Moreover, we show that, for some c > 0, there are infinitely many integers n such that φ(a) = n and σ(b) = n, each having more than n solutions. The proofs rely on the recent work of the first two author...

متن کامل

Common Values of the Arithmetic Functions

We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sum-of-divisors function. This proves a 50-year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each have more than n solutions, for some c > 0. The proofs rely on the recent work of the first two authors and K...

متن کامل

Computing the inverses, their power sums, and extrema for Euler's totient and other multiplicative functions

Wepropose a generic dynamic programming algorithm for computing the inverses of a multiplicative function. We illustrate our algorithm with Euler’s totient function and the sum of k-th powers of divisors. Our approach can be further adapted for computing certain functions of the inverses, such as their quantity, the smallest/largest inverse, which may be computed faster than the inverses themse...

متن کامل

On the Composition of Some Arithmetic Functions, Ii

We study certain properties and conjuctures on the composition of the arithmetic functions σ, φ, ψ, where σ is the sum of divisors function, φ is Euler’s totient, and ψ is Dedekind’s function.

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2007