Sphenic number

In number theory, a sphenic number (from σφήνα, 'wedge') is a positive integer that is the product of three distinct prime numbers. Because there are infinitely many prime numbers, there are also infinitely many sphenic numbers. A sphenic number is a product pqr where p, q, and r are three distinct prime numbers. In other words, the sphenic numbers are the square-free 3-almost primes. The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes. The first few sphenic numbers are 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, ... the largest known sphenic number is (282,589,933 − 1) × (277,232,917 − 1) × (274,207,281 − 1). It is the product of the three largest known primes. All sphenic numbers have exactly eight divisors. If we express the sphenic number as , where p, q, and r are distinct primes, then the set of divisors of n will be: The converse does not hold. For example, 24 is not a sphenic number, but it has exactly eight divisors. All sphenic numbers are by definition squarefree, because the prime factors must be distinct. The Möbius function of any sphenic number is −1. The cyclotomic polynomials , taken over all sphenic numbers n, may contain arbitrarily large coefficients (for n a product of two primes the coefficients are or 0). Any multiple of a sphenic number (except by 1) isn't a sphenic number. This is easily provable by the multiplication process adding another prime factor, or squaring an existing factor. The first case of two consecutive sphenic integers is 230 = 2×5×23 and 231 = 3×7×11. The first case of three is 1309 = 7×11×17, 1310 = 2×5×131, and 1311 = 3×19×23. There is no case of more than three, because every fourth consecutive positive integer is divisible by 4 = 2×2 and therefore not squarefree. The numbers 2013 (3×11×61), 2014 (2×19×53), and 2015 (5×13×31) are all sphenic. The next three consecutive sphenic years will be 2665 (5×13×41), 2666 (2×31×43) and 2667 (3×7×127) .

Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

Florian Karl Richter

A set

R\subset \mathbb{N}

is called rational if it is well approximable by finite unions of arithmetic progressions, meaning that for every

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there exists a set

B=\bigcup _{i=1}^{r}a_{i}\mathbb{N}+b_{i}

, where $a_{1},\ldots ,a_ ...

2019

A spectral refinement of the Bergelson–Host–Kra decomposition and new multiple ergodic theorems

Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

A spectral refinement of the Bergelson–Host–Kra decomposition and new multiple ergodic theorems