Infinite divisibility of Smith matrices
Infinite families of noncototients
For any positive integer let ϕ(n) be the Euler function of n. A positive integer is called a noncototient if the equation x-ϕ(x)=n has no solution x. In this note, we give a sufficient condition on a positive integer k such that the geometrical progression consists entirely of noncototients. We then use computations to detect seven such positive integers k.
Integer matrices related to Liouville's function
In this note, we construct some integer matrices with determinant equal to certain summation form of Liouville's function. Hence, it offers a possible alternative way to explore the Prime Number Theorem by means of inequalities related to matrices, provided a better estimate on the relation between the determinant of a matrix and other information such as its eigenvalues is known. Besides, we also provide some comparisons on the estimate of the lower bound of the smallest singular value. Such discussion...
Iterating the sum-of-divisors function.
Lambert series and Liouville's identities [Book]
Large and small gaps between consecutive Niven numbers.
Le déterminant de Smith et Mansion
Linear congruences and a conjecture of Bibak
We address three questions posed by K. Bibak (2020), and generalize some results of K. Bibak, D. N. Lehmer and K. G. Ramanathan on solutions of linear congruences . In particular, we obtain explicit expressions for the number of solutions, where ’s are squares modulo . In addition, we obtain expressions for the number of solutions with order restrictions or with strict order restrictions in some special cases. In these results, the expressions for the number of solutions involve Ramanujan...
Long gaps between deficient numbers
Lower bounds for the largest prime factor of an odd perfect number which is divisible by a Fermat prime.
Lucas factoriangular numbers
We show that the only Lucas numbers which are factoriangular are and .
Majoration explicite de l'ordre maximum d'un élément du groupe symétrique
Majorations de fonctions arithmétiques en moyenne sur des ensembles de faible densité
Majorations de la fonction sommatoire de la fonction de Möbius
Matrix transformations of integer sequences.
Mean values of generalized gcd-sum and lcm-sum functions.
Mellin Transforms and Fourier-Ramanujan Expansions.
Mittelwert der Eulerschen ...-Funktion und des Quadrates der Dirichletschen Teilerfunktion in algebraischen Zahlkörpern.
Monotonicity of multiplicative functions
Multiple Perfect Numbers, Mersenne Primes, and Effective Computability.