# Volume 11, 2005, Number 4

**Volume 11** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**On the structure of certain counting polynomials**

*Original research paper. Pages 1—11*

R. Valenza

Abstract

We consider a natural generalization ζ^{(k)}(*s*) = Σ*α*_{n}/*n*^{s} (*k* ≥ 2) of the Riemann zeta function that arises from a modification of its classical Euler product expansion, for the most part here concentrating on the case *k* = 2. The associated coefficients *α*_{n} correspond to a counting problem that may be addressed via a family of multivariable generating functions. Examples computed via symbolic manipulation suggest a recursive structure for these functions, which we prove. With this result in hand, the calculation of the α_{n} may be facilitated by a more efficient, double modular algorithm, as worked out in a detailed example. We conclude with some observations and questions for the case *k* > 2.

**On two new 2–Fibonacci sequences**

*Original research paper. Pages 12—16*

Krassimir T. Atanassov

**Fermat and Mersenne numbers**

*Original research paper. Pages 17—24*

J. V. Leyendekkers and A. G. Shannon

Abstract

Fermat numbers (*F*_{n} = 2^{2n} + 1) and Mersenne numbers (*M*_{m} = 2^{m} − 1), *m* odd, are compared on the basis of integer structure, using the modular rings Z_{4} and Z_{6}. The two numbers fall in different classes and this results in different composite row structures and different potentials for the formation of primes. The constraints on 2^{n} and the right end digits for *F*_{n} result in fewer numbers over a given range than those for *M*_{m}. This is shown with two functions, which link the two numbers and show that *F*_{n} = (2^{x2 + y2 − 1} + 1): for primes *y* = *n*, but when *n* > 4, *y* ≠ *n*.

**On the sum of equal powers of the first ***n* terms of an arbitrary arithmetic progression. II

*Original research paper. Pages 25—28*

Peter Vassilev and Mladen Vassilev-Missana

**Volume 11** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4