D ec 1 99 9 PRIME DIVISORS OF THE LAGARIAS SEQUENCE
نویسنده
چکیده
We solve a 1985 challenge problem posed by Lagarias [5] by determining, under GRH, the density of the set of prime numbers that occur as divisor of some term of the sequence {xn}∞ n=1 defined by the linear recurrence xn+1 = xn + xn−1 and the initial values x0 = 3 and x1 = 1. This is the first example of a ‘non-torsion’ second order recurrent sequence with irreducible recurrence relation for which we can determine the associated density of prime divisors.
منابع مشابه
σ-sporadic prime ideals and superficial elements
Let $A$ be a Noetherian ring, $I$ be an ideal of $A$ and $sigma$ be a semi-prime operation, different from the identity map on the set of all ideals of $A$. Results of Essan proved that the sets of associated prime ideals of $sigma(I^n)$, which denoted by $Ass(A/sigma(I^n))$, stabilize to $A_{sigma}(I)$. We give some properties of the sets $S^{sigma}_{n}(I)=Ass(A/sigma(I^n))setminus A_{sigma}(I...
متن کاملA remark on the means of the number of divisors
We obtain the asymptotic expansion of the sequence with general term $frac{A_n}{G_n}$, where $A_n$ and $G_n$ are the arithmetic and geometric means of the numbers $d(1),d(2),dots,d(n)$, with $d(n)$ denoting the number of positive divisors of $n$. Also, we obtain some explicit bounds concerning $G_n$ and $frac{A_n}{G_n}$.
متن کاملPrimitive Prime Divisors in Zero Orbits of Polynomials
Let (bn) = (b1, b2, . . . ) be a sequence of integers. A primitive prime divisor of a term bk is a prime which divides bk but does not divide any of the previous terms of the sequence. A zero orbit of a polynomial φ(z) is a sequence of integers (cn) where the n-th term is the n-th iterate of φ at 0. We consider primitive prime divisors of zero orbits of polynomials. In this note, we show that f...
متن کاملA Few New Facts about the EKG Sequence
The EKG sequence is defined as follows: a1 = 1, a2 = 2 and an is the smallest natural number satisfying gcd(an−1, an) > 1 not already in the sequence. The sequence was previously investigated by Lagarias, Rains and Sloane. In particular, we know that (an) is a permutation of the natural numbers and that the prime numbers appear in this sequence in an increasing order. Lagarias, Rains and Sloane...
متن کاملOn the Srnarandache Irrationality Conjecture
Here is an immediate proof in the following cases: 1. a(n) = n: 2. a(n) = d(n) =number of divisors of n; 3. a(n) = w(n) =number of distinct prime divisors of n: 4. a(n) = D(n) =number of total prime divisors of n (that is. counted with repetitions): 5. a(n) = dJ(n) =the Euler function of n: 6. a( n) = cr( n) =the sum of the divisors of n; 7. a(n) = Pn =the nth prime: 8. a(n) = 71(n) =the number...
متن کاملذخیره 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 2008