Translation invariance and finite additivity in a probability measure on the natural numbers by Robert Gardner, Robert Price

“…Inspired by the two envelopes exchange paradox, a finitely additive probability measure m on the natural numbers is introduced. The measure is uniform in the sense that m({i})=m({j}) for all i,j∈ℕ. …”
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An introduction of F-graphs, a graph-theoretic representation of natural numbers by E. J. Farrell

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Sum of the Hurwitz-Lerch Zeta Function over Natural Numbers: Derivation and Evaluation by Robert Reynolds, Allan Stauffer

“…We consider a Hurwitz-Lerch zeta function Φs,z,a sum over the natural numbers. We provide an analytically continued closed form solution for this sum in terms of the addition of Hurwitz-Lerch zeta functions. …”
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Lucas numbers of the form PX2, where P is prime by Neville Robbins

“…Let Ln denote the nth Lucas number, where n is a natural number. Using elementary techniques, we find all solutions of the equation: Ln=px2 where p is prime and p<1000.…”
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Graceful numbers by Kiran R. Bhutani, Alexander B. Levin

“…We construct a labeled graph D(n) that reflects the structure of divisors of a given natural number n. We define the concept of graceful numbers in terms of this associated graph and find the general form of such a number. …”
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More on lines in Euclidean Ramsey theory by Conlon, David, Wu, Yu-Han

“…Answering a question raised by Fox and the first author and independently by Arman and Tsaturian, we show that there is a natural number $m$ and a red/blue-colouring of $\mathbb{E}^n$ for every $n$ that contains no red copy of $\ell _3$ and no blue copy of $\ell _m$.…”
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Arithmetic functions associated with infinitary divisors of an integer by Graeme L. Cohen, Peter Hagis

“…The infinitary divisors of a natural number n are the products of its divisors of the form pyα2α, where py is a prime-power component of n and ∑αyα2α (where yα=0 or 1) is the binary representation of y. …”
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Some special cases on Stolarsky’s means by Cesare Palmisani

“…In the subfamily of SM having a natural number as parameter, we geometrically interpret one of them in particular as a real elliptic cone. …”
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Commutator Length of Finitely Generated Linear Groups by Mahboubeh Alizadeh Sanati

“…The commutator length “cl(𝐺)” of a group 𝐺 is the least natural number 𝑐 such that every element of the derived subgroup of 𝐺 is a product of 𝑐 commutators. …”
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The trajectory-coherent approximation and the system of moments for the Hartree type equation by V. V. Belov, A. Yu. Trifonov, A. V. Shapovalov

“…The formal solutions of the Cauchy problem for this equation, asymptotic in small parameter ℏ (ℏ→0), are constructed with a power accuracy of O(ℏ N/2), where N is any natural number. In constructing the semiclassically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for centered moments) is essentially used. …”
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Unique gating strategy identifier based on the Prime Population System and Gödel Numbers by Antonio Cosma

“…The unique identifiers hold since according to the fundamental theorem of arithmetic, every natural number other than one either is a prime or is a product of prime numbers, and every nonprime number can be factored into a product of prime in only one way. …”
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Some results concerning exponential divisors by Peter Hagis

“…If the natural number n has the canonical form p1a1p2a2…prar then d=p1b1p2b2…prbr is said to be an exponential divisor of n if bi|ai for i=1,2,…,r. …”
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Consecutive Power Occurrences in Sturmian Words by Bell, Jason, Schulz, Chris, Shallit, Jeffrey

“…We then give a general result showing that for every $e \in [1,(5+\sqrt{5})/2)$ there is a natural number $N$, depending only on $e$, such that every Sturmian word has the property that the distance between consecutive ending positions of $e$-powers occurring in the word is uniformly bounded by $N$.…”
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Radio and Radial Radio Numbers of Certain Sunflower Extended Graphs by Mohammed K. A. Kaabar, Kins Yenoke

“…For any integer k, 1≤k≤d, radio k−coloring of G is an assignment φ of color (positive integer) to the vertices of G such that da,b+φa−φb≥1+k, ∀a,b∈VG, where da,b is the distance between a and b in G. The biggest natural number in the range of φ is called the radio k−chromatic number of G, and it is symbolized by rckφ. …”
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A topological property of β(N) by Anastase Nakassis

“…In this paper we prove that the Stone-Cech-compactification of the natural numbers does not admit a countable infinite decomposition into subsets homeomorphic to each other and to the said compactification.…”
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On the Probabilistic Proof of the Convergence of the Collatz Conjecture by Kamal Barghout

“…A new approach towards probabilistic proof of the convergence of the Collatz conjecture is described via identifying a sequential correlation of even natural numbers by divisions by 2 that follows a recurrent pattern of the form x,1,x,1…, where x represents divisions by 2 more than once. …”
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On the K-th extension of the Sieve of eratosthenes by Antonio R. Quesada

“…For a given k we let the initial set Sk consists of natural numbers relatively prime to the first k primes, and find the rules governing the positions of the multiples of the remaining elements.…”
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Analytic Erdös-Turán conjectures and Erdös-Fuchs theorem by L. Haddad, C. Helou, J. Pihko

“…The sequences of such coefficients have a lot of similarity with sequences of natural numbers considered in additive number theory. …”
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The chromatic sum of a graph: history and recent developments by Ewa Kubicka

“…The chromatic sum of a graph is the smallest sum of colors among all proper colorings with natural numbers. The strength of a graph is the minimum number of colors necessary to obtain its chromatic sum. …”
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Finding product sets in some classes of amenable groups by Dimitrios Charamaras, Andreas Mountakis

“…In [15], using methods from ergodic theory, a longstanding conjecture of Erdős (see [5, Page 305]) about sumsets in large subsets of the natural numbers was resolved. In this paper, we extend this result to several important classes of amenable groups, including all finitely generated virtually nilpotent groups and all abelian groups $(G,+)$ with the property that the subgroup $2G := \{g+g : g\in G\}$ has finite index. …”
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