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Equations relating factors in decompositions into factors of some family of plane triangulations, and applications (with an appendix by Andrzej Schinzel)

Jan Florek (2015)

Colloquium Mathematicae

Let be the family of all 2-connected plane triangulations with vertices of degree three or six. Grünbaum and Motzkin proved (in dual terms) that every graph P ∈ has a decomposition into factors P₀, P₁, P₂ (indexed by elements of the cyclic group Q = 0,1,2) such that every factor P q consists of two induced paths of the same length M(q), and K(q) - 1 induced cycles of the same length 2M(q). For q ∈ Q, we define an integer S⁺(q) such that the vector (K(q),M(q),S⁺(q)) determines the graph P (if P is...

Expansions of binary recurrences in the additive base formed by the number of divisors of the factorial

Florian Luca, Augustine O. Munagi (2014)

Colloquium Mathematicae

We note that every positive integer N has a representation as a sum of distinct members of the sequence d ( n ! ) n ≥ 1 , where d(m) is the number of divisors of m. When N is a member of a binary recurrence u = u ₙ n ≥ 1 satisfying some mild technical conditions, we show that the number of such summands tends to infinity with n at a rate of at least c₁logn/loglogn for some positive constant c₁. We also compute all the Fibonacci numbers of the form d(m!) and d(m₁!) + d(m₂)! for some positive integers m,m₁,m₂.

Extensions of Three Theorems of Nagell

A. Schinzel (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

Three theorems of Nagell of 1923 concerning integer values of certain sums of fractions are extended.

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