An arithmetic function
نویسندگان
چکیده
منابع مشابه
The convolution inverse of an arithmetic function
Good, but does an inverse g of f have to exist? A necessary condition is that f(1) 6= 0. Indeed, if g is the inverse of f , then 1 = I(1) = (f ∗ g)(1) = f(1)g(1). We now show that this necessary condition is also sufficient. We assume that f(1) 6= 0 and we try and solve for g. What this means is that we are solving for infinitely many unknowns: g(1), g(2), . . . . From the above, we see that th...
متن کاملAn arithmetic function arising from Carmichael’s conjecture
Let φ denote Euler’s totient function. A century-old conjecture of Carmichael asserts that for every n, the equation φ(n) = φ(m) has a solution m 6= n. This suggests defining F (n) as the number of solutions m to the equation φ(n) = φ(m). (So Carmichael’s conjecture asserts that F (n) ≥ 2 always.) Results on F are scattered throughout the literature. For example, Sierpiński conjectured, and For...
متن کاملAn Arithmetic Formula for the Partition Function
Although there is a vast literature on the properties of p(n), typically motivated by work of Ramanujan, some of the simplest questions remain open. For example, little is known about p(n) modulo 2 and 3. Most results concerning the congruence properties of p(n) have been proved using properties of (1.1). Theorems typically depend on q-series identities, the theory of modular equations, or the ...
متن کاملOn an arithmetic function considered by Pillai
For every positive integer n let p(n) be the largest prime number p ≤ n. Given a positive integer n = n1, we study the positive integer r = R(n) such that if we define recursively ni+1 = ni − p(ni) for i ≥ 1, then nr is a prime or 1. We obtain upper bounds for R(n) as well as an estimate for the set of n whose R(n) takes on a fixed value k.
متن کاملFunction Interval Arithmetic
We propose an arithmetic of function intervals as a basis for convenient rigorous numerical computation. Function intervals can be used as mathematical objects in their own right or as enclosures of functions over the reals. We present two areas of application of function interval arithmetic and associated software that implements the arithmetic: (1) Validated ordinary differential equation sol...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Bulletin of the American Mathematical Society
سال: 1937
ISSN: 0002-9904
DOI: 10.1090/s0002-9904-1937-06534-7