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Carmichael numbers composed of primes from a Beatty sequence

William D. Banks, Aaron M. Yeager (2011)

Colloquium Mathematicae

Let α,β ∈ ℝ be fixed with α > 1, and suppose that α is irrational and of finite type. We show that there are infinitely many Carmichael numbers composed solely of primes from the non-homogeneous Beatty sequence α , β = ( α n + β ) n = 1 . We conjecture that the same result holds true when α is an irrational number of infinite type.

Composite positive integers whose sum of prime factors is prime

Florian Luca, Damon Moodley (2020)

Archivum Mathematicum

In this note, we show that the counting function of the number of composite positive integers n x such that β ( n ) = p n p is a prime is of order of magnitude at least x / ( log x ) 3 and at most x / log x .

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