Lower bounds for numbers of ABC-hits

By an ABC-hit, we mean a triple (a,b,c) of relatively prime positive integers such that a+b=c and rad(abc)<c. Denote by N(X) the number of ABC-hits (a,b,c) with c?X. In this paper we discuss lower bounds for N(X). In particular we prove that for every ?>0 and X large enough N(X)?exp((logX)^1/2−?).     

B-free numbers in short arithmetic progressions

Given a sequence B of relatively prime positive integers with the sum of inverses finite, we investigate the problem of finding B-free numbers in short arithmetic progressions.     

Sums of triangular numbers from the Frobenius determinant

We show that the denominator formula for the strange series of affine superalgebras, conjectured by Kac and Wakimoto and proved by Zagier, follows from a classical determinant evaluation of Frobenius. As a limit case, we obtain exact formulas for the number of representations of an arbitrary number as a sum of 4m²/d triangles, whenever d|2m, and 4m(m+1)/d triangles, when d|2m or d|2m+2. This extends recent results of Getz and Mahlburg, Milne, and Zagier.     

On Delannoy numbers and Schröder numbers

The nth Delannoy number and the nth Schröder number given by

     

Practical numbers in Lucas sequences

Abstract

A practical number is a positive integer n such that all the positive integers m ≤ n can be written as a sum of distinct divisors of n. Let (u_n)_n≥0 be the Lucas sequence satisfying u₀ = 0, u₁ = 1, and u_n+2 = au_n+1 + bu_n for all integers n ≥ 0, where a and b are fixed nonzero integers. Assume a(b + 1) even and a² + 4b > 0. Also, let be the set of all positive integers n such that |u_n| is a practical number. Melfi proved that is infinite. We improve this result by showing that #(x) ? x/log x for all x ≥ 2, where the implied constant depends on a and b. We also pose some open questions regarding .     

Multiperfect numbers with identical digits

Let g?2. A natural number N is called a repdigit in base g if all of the digits in its base g expansion are equal, i.e., if for some m?1 and some D∈{1,2,…,g−1}. We call N perfect if σ(N)=2N, where σ denotes the usual sum-of-divisors function. More generally, we call N multiperfect if σ(N) is a proper multiple of N. The second author recently showed that for each fixed g?2, there are finitely many repdigit perfect numbers in base g, and that when g=10, the only example is N=6. We prove the same results for repdigit multiperfect numbers.     

On the distribution of square-full and cube-full numbers

There are many results on the distribution of square-full and cube-full numbers. In this article the distribution of these numbers are studied in more detail. Suchk-full numbers (k=2,3) are considered which are at the same time 1-free (1k+2). At first an asymptotic result is given for the numberN _k,1(x) ofk-full and 1-free numbers not exceedingx. Then the distribution of these numbers in short intervals is investigated. We obtain different estimations of the differenceN _k,1(x+h)–N_k,1(x) in the casesk=2, 1=4,5,6,7,18 andk=3, 1=5,6,7, 18.     

Davenport constant for finite abelian groups

For a finite abelian group G, we investigate the length of a sequence of elements of G that is guaranteed to have a subsequence with product identity of G. In particular, we obtain a bound on the length which takes into account the repetitions of elements of the sequence, the rank and the invariant factors of G. Consequently, we see that there are plenty of such sequences whose length could be much shorter than the best known upper bound for the Davenport constant of G, which is the least integer s such that any sequence of length s in G necessarily contains a subsequence with product identity. We also show that the Davenport constant for the multiplicative group of reduced residue classes modulo n is comparatively large with respect to the order of the group, which is φ(n),when n is in certain thin subsets of positive integers. This is done by studying the Carmichael’s lambda function, defined as the maximal multiplicative order of any reduced residue modulo n, along these subsets.     

Gaps in semigroups

In this paper we investigate the behaviour of the gaps in numerical semigroups. We will give an explicit formula for the ith gap of a semigroup generated by k+1 consecutive integers (generalizing a result due to Brauer) as well as for a special numerical semigroup of three generators. It is also proved that the number of gaps of the numerical semigroup generated by integers p and q with g.c.d.(p,q)=1, in the interval [pq-(k+1)(p+q),…,pq-k(p+q)] is equals to

     

Further results on the euler and Genocchi numbers

Summary We characterize the ordinary generating functions of the Genocchi and median Genocchi numbers as unique solutions of some functional equations and give a direct algebraic proof of several continued fraction expansions for these functions. New relations between these numbers are also obtained.     

Density of sets of natural numbers and the Lévy group

Let N denote the set of positive integers. The asymptotic density of the set A⊆N is d(A)=limn→∞|A∩[1,n]|/n, if this limit exists. Let AD denote the set of all sets of positive integers that have asymptotic density, and let SN denote the set of all permutations of the positive integers N. The group L^? consists of all permutations f∈SN such that A∈AD if and only if f(A)∈AD, and the group L^* consists of all permutations f∈L^? such that d(f(A))=d(A) for all A∈AD. Let be a one-to-one function such that d(f(N))=1 and, if A∈AD, then f(A)∈AD. It is proved that f must also preserve density, that is, d(f(A))=d(A) for all A∈AD. Thus, the groups L^? and L^* coincide.     

Sums of products of Kronecker's double series

Closed expressions are obtained for sums of products of Kronecker's double series of the form , where the summation ranges over all nonnegative integers j₁,…,jN with j₁+?+jN=n. Corresponding results are derived for functions which are an elliptic analogue of the periodic Euler polynomials. As corollaries, we reproduce the formulas for sums of products of Bernoulli numbers, Bernoulli polynomials, Euler numbers, and Euler polynomials, which were given by K. Dilcher.     

A remark on sociable numbers of odd order

Write s(n) for the sum of the proper divisors of the natural number n. We call n sociable if the sequence n, s(n), s(s(n)), … is purely periodic; the period is then called the order of sociability of n. The ancients initiated the study of order 1 sociables (perfect numbers) and order 2 sociables (amicable numbers), and investigations into higher-order sociable numbers began at the end of the 19th century. We show that if k is odd and fixed, then the number of sociable n?x of order k is bounded by as x→∞. This improves on the previously best-known bound of , due to Kobayashi, Pollack, and Pomerance.     

Maximal Collections of Intersecting Arithmetic Progressions

Let N _t (k) be the maximum number of k-term arithmetic progressions of real numbers, any two of which have t points in common. We determine N ₂(k) for prime k and all large k, and give upper and lower bounds for N _t (k) when t 3.* Research supported in part by NSF grant DMS-0070618.     

Odd multiperfect numbers of abundancy 4

Euler's structure theorem for any odd perfect number is extended to odd multiperfect numbers of abundancy power of 2. In addition, conditions are found for classes of odd numbers not to be 4-perfect: some types of cube, some numbers divisible by 9 as the maximum power of 3, and numbers where 2 is the maximum even prime power.     

Supersequences, rearrangements of sequences, and the spectrum of bases in additive number theory

The set of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be represented as the sum of h elements of A. If an∼αnh for some real number α>0, then α is called an additive eigenvalue of order h. The additive spectrum of order h is the set N(h) consisting of all additive eigenvalues of order h. It is proved that there is a positive number ηh?1/h! such that N(h)=(0,ηh) or N(h)=(0,ηh]. The proof uses results about the construction of supersequences of sequences with prescribed asymptotic growth, and also about the asymptotics of rearrangements of infinite sequences. For example, it is proved that there does not exist a strictly increasing sequence of integers such that bn∼n² and B contains a subsequence such that bnk∼k³.     

Multiperfect numbers on lines of the Pascal triangle

A number n is said to be multiperfect (or multiply perfect) if n divides its sum of divisors σ(n). In this paper, we study the multiperfect numbers on straight lines through the Pascal triangle. Except for the lines parallel to the edges, we show that all other lines through the Pascal triangle contain at most finitely many multiperfect numbers. We also study the distribution of the numbers σ(n)/n whenever the positive integer n ranges through the binomial coefficients on a fixed line through the Pascal triangle.