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In this paper, we give a refined form of a recent factorization of Lambert series. This result allows us to prove new connections between partitions and divisors of positive integers, such as a new formula for the number of divisors of a positive integer as a convolution. Three recurrence relations for computing the number of partitions of a positive integer into distinct parts are rediscovered in this context.

In this paper, we generalize the result obtained by G. Bareikis and E. Manstavicius [On the DDT theorem, Acta Arith., 126(2):155–168, 2007] for a sequence of integers from a probabilistic point of view. As an application, we give the distribution law of divisors of an integer n that can be written as the sum of two squares.

The q -binomial coefficients are specializations of the elementary symmetric functions. In this paper, we use this fact to give a new expression for the generating function of the number of divisors. As corollaries, we obtained new connections between partitions and divisors.

We compute divisors class groups of singular surfaces. Most notably we produce an exact sequence that relates the Cartier divisors and almost Cartier divisors of a surface to the those of its normalization. This generalizes Hartshorne's theor em for the cubic ruled surface in P 3 . We apply these results to limit the possible curves that can be s et-theoretic complete intersection in P 3 in characteristic zero. On a nonsingular variety, the study of divisors and linear systems is classical. In fact the entire theory of curves and surfaces is dependent on this study of codimension one subvarieties and the linear and algebraic families in which they move. This theory has been generalized in two directions: the Weil divisors on a normal variety, taking codimension one subvarieties as prime divisors; and the Cartier divisors on an arbitrary scheme, based on locally principal codimension one subschemes. Most of the literature both in algebraic geometry and commutative algebra up to now has been limited to these kinds of divisors. More recently there have been good reasons to consider divisors on non-normal varieties. Jaffe (9) introduced the notion of an almost Cartier divisor, which is locally principal off a subset of codimension two. A theory of generalized divisors was proposed on curves in (14), and extended to any dimension in (15). The latter paper gave a complete description of the generalized divisors on the ruled cubic surface in P 3 . In this paper we extend that analysis to an arbitrary integra l surface X, explaining the group APicX of linear equivalence classes of almost Cartier divisors on X in terms of the Picard group of the normalization S of X and certain local data at the singular points of X. We apply these results to give limitations on the possible curves that can b e set-theoretic compete intersections in P 3 in characteristic zero In section 2 we explain our basic set-up, comparing divisors on a variety X to its normalization S. In Section 3 we prove a local isomorphism that computes the group of almost Cartier divisors at a singular point of X in terms of the Cartier divisors along the curve of singulari ties and its inverse image in the normalization. In Section 4 we derive some global exact sequences for the groups PicX, APicX, and PicS, which generalize the results of (15, §6) to arbitrary surfaces These results are particularly transparent for surfaces wi th ordinary singularities, meaning a double curve with a finite number of pinch points and triple points.

We study the number of divisors in residue classes modulo m and prove, for example, that the exact equidistribution holds for almost all natural numbers coprime to m in the sense of natural density if and only if m = 2kp1p2…ps, where k and s are non-negative integers and pj are distinct Fermat primes. We also provide a general and exact lower bound for the proportion of divisors in the residue class 1 mod m. The same combinatorial technique using Davenport's constant leads to exact lower bounds for the number of representations of a natural number by a given binary quadratic form with a negative discriminant.

Let N(a1, …, a4; n) denote the number of representations of an integer n by the form [Formula: see text]. In this paper we derive formulae for N(1, 1, 1, 2; n) and N(1, 2, 2, 2; n). These formulae are given in terms of σ3(n).

We investigate a new problem related to the multiplicative structure of integers. We introduce a new function H1, 1(x, y, z1, z2), the number of positive integers n ≤ x having exactly one divisor in interval (y, z1] and one in (z1, z2]. Our aim is to estimate its order of magnitude under some general conditions on y, z1 and z2.

In this note, we will give intersection numbers for two simple 2-fold ( 3 n , n , 3 ) group divisible designs (GDD). More precisely, we note that there exists two simple 2-fold ( 3 n , n , 3 ) GDDs which intersect in precisely k ∈ { 0 , 1 , 2 , … , 2 n 2 } \ { 2 n 2 − 1 , 2 n 2 − 2 , 2 n 2 − 3 , 2 n 2 − 5 } triples for n ⩾ 5 . There are some exceptions for n = 3 , 4 .

Given a quaternion integer α whose norm is divisible by a natural number m , does there exist a quaternion integer β of norm m dividing α on both the left and right? This problem is a case of the “metacommutation problem”, which asks generally for relationships between the many different factorizations of a given integral quaternion. In this paper, we give necessary and sufficient conditions on primitive α of odd norm to ensure the existence of common left- and right-hand divisors, and we characterize the non-trivial sets of such divisors.

In this note, we prove that the Diophantine equation 2 m + n x 2 = y n in positive integers x, y, m, n has the only solution ( x , y , m , n ) = ( 21 , 11 , 3 , 3 ) with n > 1 and gcd ( n x , y ) = 1 . In fact, for n = 3 , 15 , we transform the above equation into several elliptic curves for which we determine all their {2}-integer points. For n ≠ 3 , 15 , we apply the result of Yu.F. Bilu, G. Hanrot and P.M. Voutier about primitive divisors of Lehmer sequences.

The notion of generalized divisors on schemes is introduced by Hartshorne. It is shown that there exists a bijection between the set of all generalized divisors on a scheme X and the set of all reflexive coherent 𝒪 X -modules which are locally free of rank one at generic points. This bijection, corresponds Cartier divisors to the set of all locally free sheaves of rank one. Our aim in this article is to study the class of generalized divisors that maps to totally reflexive coherent 𝒪 X -modules, under this correspondence. We investigate this class of divisors, that will be called Gorenstein divisors, both over schemes and also over commutative noetherian rings. We show that this class of divisors has usual properties and fits well in the hierarchy of divisors that already exists in the literature.

Let σ j ( n ) = ∑ d | n d j be the sum of divisors function, and let I be the identity function. When considering only one input variable n , we show that the set of functions { σ i } i = 0 ∞ ∪ { I } is algebraically independent. With two input variables, we give a non-trivial identity involving the sum of divisors function, prove its uniqueness, and use it to prove that any perfect number n must have the form n = r σ ( r ) / ( 2 r − σ ( r ) ) , with some restrictions on r . This generalizes the known forms for both even and odd perfect numbers.

An asymptotic formula is given for the number of integers n ≤ x which do not have divisors in a fixed arithmetical progression. This extends a previous result of Banks et al. (Forum Math 20:1005–1037, 2008) who considered the case of progressions with prime difference.

Let ? stand for the Euler function. Given a positive integer n, let ?(n) stand for the sum of the positive divisors of n and let ?(n) be the number of divisors of n. We obtain an asymptotic estimate for the counting function of the set {n : gcd (?(n), ?(n)) = gcd(?(n), ?(n)) = 1}. Moreover, setting l(n) : = gcd(?(n), ?(n+1)), we provide an asymptotic estimate for the size of #{n ? x: l(n) = 1}.

An integer t is a twin prime (see [5] or [6]), if t is a prime number ≥ 3 and if t − 2 or t + 2 is also a prime number ≥ 3 . Example. 1000000000061 and 1000000000063 are twin primes (see [6]). It is conjectured that there are infinitely many twin primes. The notion of a, friendly number (see [2] or [3] or [7] or [8] or [9] or [10]) is based on the idea that a human friend is a kind of alter ego. Indeed, Pythagoras wrote (see [8]): A friend is the other I , such as are 220 and 284 . These numbers have a special mathematical property: each is equal to the sum of the other’s proper divisors (divisors other than the number itself). The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 , and 110, and they sum to 284; the proper divisors of 284 are 1, 2, 4, 71 , and 142, and they sum to 220. So {220, 284} is called a pair of friendly numbers [[note {17296, 18416} is also a pair of friendly numbers (see [7] or [8])]]. More precisely, we say that a number a′ is a friendly number, if there exi...

Given an integer q≥2, we say that a positive integer is a q-Niven number if it is divisible by the sum of its digits in base q. Given an arbitrary integer r∈[2,2q], we say that (n,n+1,…,n+r−1) is a q-Niven r-tuple if each number n+i, for i=0,1,…,r−1, is a q-Niven number. We show that there exists a positive constant c=c(q,r) such that the number of q-Niven r-tuples whose leading component is

The aim of the present paper is to characterize prime numbers of the form n = x 2 + (x + 1)2 and to obtain certain proper divisors of composite numbers of the same form, i.e. divisors d of n such that 1 < d < n.

In this paper, we study various arithmetic properties of the sequence (an)n≥1 satisfying the recurrence relation an = nan–1 + 1, n = 2, 3,..., with the initial term a1 = 0. In particular, we estimate the number of solutions of various congruences with this sequence and the number of distinct prime divisors of its first N terms.

It is known that the sequence 1,2,1,1,2,2,2,1,1,2,1,1,2,1,1,2,2,… of lengths of blocks of identical symbols in the Thue–Morse sequence has several extremal properties among all non-periodic sequences of the symbols 1 and 2. Its generating function W(x) is equal to ∏k=1∞(1+x(2k+(-1)k-1)/3). In terms of combinatorics on words, for any given x∈(0,1) and ε>0, we prove that every non-periodic word of an alphabet {1,2} has a suffix s whose generating function S(x) satisfies the inequality xS(-x)>1-W(-x)-ε. Using this, we prove several bounds for the largest and the smallest limit points of the sequence of fractional parts {ξbn}, n=0,1,2,…, where b

It is proved that the set of those natural numbers which cannot be written as n-Ω(n) is of positive lower density. Here Ω(n) is the number of the prime power divisors of n. This is a refinement of a theorem of F. Luca.

Let g and m be two positive integers, and let F be a polynomial with integer coefficients. We show that the recurrent sequence x0 = g, xn = x n−1 n + F(n), n = 1, 2, 3,…, is periodic modulo m. Then a special case, with F(z) = 1 and with m = p > 2 being a prime number, is considered. We show, for instance, that the sequence x0 = 2, xn = x n−1 n + 1, n = 1, 2, 3, …, has infinitely many elements divisible by every prime number p which is less than or equal to 211 except for three prime numbers p = 23, 47, 167 that do not divide xn. These recurrent sequences are related to the construction of transcendental numbers ζ for which the sequences [ζn!], n = 1, 2, 3, …, have some nice divisibility properties. Bibliography: 18 titles.

A Dedekind-finite set is said to be divisible by a natural number n if it can be partitioned into pieces of size n. We study several aspects of this notion, as well as the stronger notion of being partitionable into n pieces of equal size. Among our results are that the divisors of a Dedekind-finite set can consistently be any set of natural numbers (containing 1 but not 0), that a Dedekind-finite power of 2 cannot be divisible by 3, and that a Dedekind-finite set can be congruent modulo 3, to all of 0, 1, and 2 simultaneously. (In these results, 2 and 3 serve as typical examples; the full results are more general.)

We consider the following integer feasibility problem: Given positive integer numbers a0, a1,…,an, with gcd(a1,…,an) = 1 and a = (a1,…,an), does there exist a vector x ∈ ℤn≥0 satisfying a x = a0? We prove that if the coefficients a1,…,an have a certain decomposable structure, then the Frobenius number associated with a1,…,an, i.e., the largest value of a0 for which a x = a0 does not have a nonnegative integer solution, is close to a known upper bound. In the instances we consider, we take a0 to be the Frobenius number. Furthermore, we show that the decomposable structure of a1,…,an makes the solution of a lattice reformulation of our problem almost trivial, since the number of lattice hyperplanes that intersect the polytope resulting from the reformulation in the direction of the last coordinate is going to be very small. For branch-and-bound such instances are difficult to solve, since they are infeasible and have large values of a0/ai, 1 ≤ i ≤ n. We illustrate our results by some computational examples.

We present analogies between Diophantine conditions appearing in the theory of Small Divisors and classical Transcendental Number Theory. Let K be a number field. Using Bertrand's postulate, we give a simple proof that $e$ is transcendental over Liouville fields K$(\theta)$ where $\theta $ is a Liouville number with explicit very good rational approximations. The result extends to any Liouville field K$(\Theta )$ generated by a family $\Theta$ of Liouville numbers satisfying a Diophantine condition (the transcendence degree can be uncountable). This Diophantine condition is similar to the one appearing in Moser's theorem of simultanneous linearization of commuting holomorphic germs.

For a positive integer j we look at the parity of the number of divisors of n that are at most j, proving that for large j, the count is even for most values of n.

We study holomorphic curves in ann-dimensional complex manifold on which a family of divisors parametrized by anm-dimensional compact complex manifold is given. If, for a given sequence of such curves, their areas (in the induced metric) monotonically tend to infinity, then for every divisor one can define adefect characterizing the deviation of the frequency at which this sequence intersects the divisor from the average frequency (over the set of all divisors). It turns out that, as well as in the classical multidimensional case, the set of divisors with positive defect is very rare. (We estimate how rare it is.) Moreover, the defect of almost all divisors belonging to a linear subsystem is equal to the mean value of the defect over the subsystem, and for all divisors in the subsystem (without any exception) the defect is not less than this mean value.

We prove that every odd perfect number N has at least 420 distinct prime divisors, and that N is greater than 1,9 × 102550.

The asymptotics of sums of the form Στ(¦bn−a¦) (summation overn

For each positive integer $n$, let $s(n)$ denote the sum of the proper divisors of $n$. If $s(n)>0$, put $s_{2}(n)=s(s(n))$, and define the higher iterates $s_{k}(n)$ similarly. In 1976, Erdős proved the following theorem: For each $\delta>0$ and each integer $K\geq2$, we have ¶ \[-\delta

Let ω(n) denote the number of distinct prime divisors of n; that is, $$\displaystyle{\omega (n) =\sum _{p\vert n}1.}$$ Thus, for example, ω(1) = 0, ω(2) = 1, ω(9) = 1, ω(60) = 3. The values of ω(n) obviously fluctuate wildly as n → ∞, since ω(p) = 1, for every prime p.

Let D 1 , D 2 be positive integers with 2∤ D 1 D 2 and gcd( D 1 , D 2 )=1. Let p be a prime with p ∤ D 1 D 2 . In this note we prove that the generalized Ramanujan–Nagell equation D 1 x 2 + D 2 =4 p n has at most two positive integer solutions ( x , n ) except ( D 1 , D 2 , p )=(1, 7, 2), (3, 5, 2), (1, 11, 3), and (1, 19, 5).

We present several methods to construct or identify families of free divisors such as those annihilated by many Euler vector fields, including binomial free divisors, or divisors with triangular discriminant matrix. We show how to create families of quasihomogeneous free divisors through the chain rule or by extending them into the tangent bundle. We also discuss whether general divisors can be extended to free ones by adding components and show that adding a normal crossing divisor to a smooth one will not succeed.

Let [math] be the sum of the divisors of [math] . Although much attention has been paid to the possible values of [math] (the sum of proper divisors), comparatively little work has been done on the possible values of [math] . Here we present some theoretical and computational results on these values. In particular, we exhibit some infinite and possibly infinite families of integers that appear in the image of [math] . We also find computationally all values of [math] for which [math] is odd, and we present some data from our computations. At the end of this paper, we present some conjectures suggested by our computational work.

Let d(n) denote the number of positive divisors of n, and let f(x) be a polynomial in x with integer coefficients, irreducible over ℤ. Erdös[3] showed that there exist constants λ1, λ2 (depending on f) such that

Let d(m) be the number of divisors of the positive integer m. Here, we show that if n !∈ {3, 5}, then d(n!) is a divisor of n!. We also show that the only positive integers n such that d(Fn) divides Fn, where Fn is the nth Fibonacci number, are n ∈ {1, 2, 3, 6, 24, 48}.

In this note we prove that the difference divisors associated with special cycles on unitary Rapoport-Zink spaces of signature (1,n-1) in the unramified case are always regular., Comment: 3 pages

We determine asymptotically the maximal order of log d(d(n)), where d(n) is the number of positive divisors of n. This solves a problem first put forth by Ramanujan in 1915., Comment: 8 pages

Let l(N, r) be the minimum number of terms needed to express r as a sum of distinct divisors of N. Let l(N) = max{l(N, r) : 1 ≤ r ≤ N}. Then for Vose's sequence improving the result of M. Vose.

The notions of unitary divisor and biunitary divisor are extended in a natural fashion to give k-ary divisors, for any natural number k. We show that we may sensibly allow k to increase indefinitely, and this leads to infinitary divisors. The infinitary divisors of an integer are described in full, and applications to the obvious analogues of the classical perfect and amicable numbers and aliquot sequences are given.

For a non-zero integer n, let d(n) denote the number of positive divisors of n. Let a, b and c be integers with a>0, and set Δ=b2−4ac. If the quadratic polynomial ax2+bx+c is irreducible over the rational numbers Q (that is, if Δ is not the square of an integer), then one hasformula hereas X→∞, for some λ depending on a, b and c (see [7]). In this paper we discuss the way in which λ depends on a, b and c, giving a precise, compact expression in terms of class numbers. This extends previous work for the case a=1, Δ

Several results involving d(n!) are obtained, where d(m) denotes the number of positive divisors of m. These include estimates for d(n!)/d((n − 1)!), d(n!), − d((n − 1)!), as well as the least number K with d((n + K)!)/d(n!) ≥ 2.

The infinitary divisors of a natural numbernare the products of its divisors of the formpyα2α, wherepyis a prime-power component ofnand∑αyα2α(whereyα=0or1) is the binary representation ofy. In this paper, we investigate the infinitary analogues of such familiar number theoretic functions as the divisor sum function, Euler's phi function and the Möbius function.

Let pn be the number of consecutive positive summands and let qn be the number of consecutive negative summands that appear in the classical Riemann’s rearrangement of the alternating harmonic series to sum a prescribed real number s. Assume that s > log2 and let x = (1=4)e 2s . It is shown that the sequence qn is constant equal to 1, and that the values of pn become stabilized: Eventually pn = bxc or pn =bxc + 1. Moreover, it is shown that x is rational if and only if the sequence pn is eventually periodic. The sequence pn is eventually constant if and only if x is integer, in which case pn =bxc for n big enough. Similar results are also true for s < log2.

Since the partitions 1 + 2 + 4 and 7 contain an odd number of summands, they are called odd partitions, whereas the other three partitions are called even. Add the smallest numbers of the odd partitions, 1 + 7= 8, and do the same for the smallest numbers of the even partitions, 1 + 2 + 3 = 6. The difference between these two sums, 8 6 = 2, is exactly the number of divisors of the prime 7. In the sequel, p(n) denotes the sum of the smallest numbers of odd partitions of n minus the smallest numbers of even partitions of n, and d(n) denotes the number of divisors of n. For small numbers n, it is easy to check that p(n) equals d(n). This is not a coincidence; we shall see that it is a general relation between the smallest numbers of partitions into unequal parts and the number of divisors.