. By replacing the numerator in the nth term of the divergent p-series and the reciprocals of primes series with the nth term of the Thue-Morse sequence, one can produce a deletion of terms in the said series, which we show remains divergent. A connection is also revealed, between the sequence (an)n >= 1 defined as the largest power of two to divide an integer n and the ordinary generating function for the Thue-Morse sequence. In addition, we provide a new elementary proof that the sequence (an)n >= 1 is square free in the context of combinatorics on words.

By applying an existing characterization for a positive integer to be represented as a sum of two cubes of positive integers, we construct an elementary proof of Ramanujan's famous result-namely, that the number 1729 is the smallest positive integer represented as a sum of two cubes in two different ways. Similarly, by applying an existing characterization for a positive integer to be represented as a difference of cubes of two positive integers, we also apply this characterization to construct the generating function for a sequence of integer ordered pairs (a(n), b(n)) not equal (a(n)', b(n)') satisfying an = an3 + b(n)(3) = a(n)'(3), b(n)'(3), which are distinct from Ramanujan's "near integer" solutions to Fermat's equation-namely, those satisfying a(n)(3) + b(n)(3) = c(n)(3) + (-1)(n).

Although no known closed-form expression exists for the Nth partial sum of the p-series, we show that a closed-form expression can at least be found for the integer part of these partial sums, where , for specific sequences of integers N that are dependent on the value of p. The closed-form expressions follow from an application of the upper and lower bounds obtained from the integral test for the partial sums of series with positive terms.

By extending the method of the author’s earlier paper, a purely recursive approach is devised for counting the total number of unfolded self-avoiding walks terminating along the line x=n, within the finite lattice strip {0,1,…,n}×{0,1,2,3} of width three. This approach yields a generating function for the sequence of numbers just described, without the need for any exact enumeration, usually entailed with an application of the transfer matrix algorithm. As a result of determining this generating function, the total number of unfolded walks within the lattice strip can then be easily computed.

A non-ergodic approach is employed to establish Benford's law for the leading digit d = 1 for the sequence of powers of two. For the sequence of powers f ng, 1 < 10=9, this method is extended to obtain a weak rst digit law by establishing Benford like lower and upper bounds to the asymptotic relative frequency of terms with a given leading digit.

Define the average path length in a connected graph G as the sum of the lengths of the shortest path between pairs of nodes is divided by the total number of pairs of nodes. Denoting the sum of the shortest path lengths between all pairs of nodes in an n x M rectangular lattice, having n > 2 fixed rows and m >= 2 variable rows by S-m, we derive a first order linear but non-homogeneous recurrence relation for Sm, from which a closed-form expression for Sm is obtained. Using this explicit expression for Sm, one can then show that the average path length within this graph must be asymptotic to, 3/D, where D is the diameter, that is, the longest shortest path.

Define the average path length in a connected graph as the sum of the length of the shortest path between all pairs of nodes, divided by the total number of pairs of nodes. Letting SN denote the sum of the shortest path lengths between all pairs of nodes in a complete m-ary tree of depth N, we derive a first-order linear but non-homogeneous recurrence relation for SN, from which a closed-form expression for SN is obtained. Using this explicit expression for SN, we show that the average path length within this graph/network is asymptotic to D - 4/(m - 1), where D is the diameter of the m-ary tree, that is, the longest shortest path. This asymptotic estimate for the average path length confirms a conjectured asymptotic estimate in the case of complete binary tree.

A new and elementary proof is presented which establishes a result of Sylvester namely, an odd perfect number must necessarily have at least three distinct prime factors. The argument depends on nothing more than a simple factorization of monic polynomials of the form 1+p+ ...+p''.

We construct a new family of normalised metrics for measuring the dissimilarity of finite sets in terms of the sizes of the sets and of their intersection. The family normalises a set-based analogue of the Minkowski metric family. It is parametrised by a real variable p≥1, is monotonic decreasing in p, equals the normalised set difference metric when p=1 and equals the normalised maximum difference metric in the limit p→∞. These metrics are suitable for comparison of finite sets in any context. Several applications to comparison of finite graphs are described.

A recursive algorithm is constructed which finds all solutions to a class of Diophantine equations connected to the problem of determining ordered n-tuples of positive integers satisfying the property that their sum is equal to their product. An examination of the use of Binary Search Trees in implementing the algorithm into a working program is given. In addition an application of the algorithm for searching possible extra exceptional values of the equal-sum-product problem is explored after demonstrating a link between these numbers and the Sophie Germain primes.

A closed-form expression is derived for the enumeration of all palindromic binary strings of length n > r having no r-runs of 1’s, in terms of the r-Fibonacci sequence. A similar closed-form expression for the number of zeros contained in all such palindromic binary strings is derived in terms of the number of zeros contained in all binary strings having no r-runs of 1’s.

Using elementary methods we show that if an integer n > 2 is an exceptional value of the equal-sum-and-product-problem, then n - 1 must be a Sophie Germain prime number. This result gives further evidence to the sparsity conjecture for the set of exceptional values of the equal-sum-and-product problem.

The number of binary strings of length n containing no substrings consisting of r consecutive ones is examined and shown to be given in terms of a well known integer sequence namely, the r-Fibonacci sequence. In addition, difference equations for the number of zeros and the total number of runs within these binary strings are derived.

By applying double and triple angle identities for hyperbolic and trigonometric cosine functions, we obtain closed-form evaluations for two families of infinite products involving nested radicals. The first group of results represents a generalization of the classic Viete infinite product expansion for 2/pi, while the second comprises variations on Viete type infinite products and infinite products involving nested square roots of 2. In addition, specific examples of Viete type infinite product expansions are presented for such numbers as 3 root 3/2 pi and 3/pi.

An enumeration formula for counting the number of partitions of an integer 71 > 1 hewing parts in