On the smallest abundant number not divisible by the first $k$ primes

Iannucci, Douglas E.

doi:10.36045/bbms/1113318127

Cited by 4 publications

(7 citation statements)

References 1 publication

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“…Then i 0 ≤ a; if not, by (31), 2 a+1 | n which is impossible. But now, again by (31), b i0 2 i0 | n which contradicts (29).…”

Section: Unique Minimal Subset (Proof Of Theorem A)mentioning

confidence: 91%

“…In [8], Bessel-Hagen asked whether the set of abundant numbers has asymptotic density and the positive answer was given independently by Davenport [13], Chowla [11] and Erdös [20]. Nowadays, abundant numbers are still of a certain interest in number theory (see, e.g., the recent works [31,32,35]).…”

Section: Motivationmentioning

confidence: 99%

“…Remark 12.5. Another way to prove the above lemma is this is to use the algorithm presented in [31], outputting the smallest abundant number not divisible by the first k primes.…”

Section: Taut Case Revisitedmentioning

confidence: 99%

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$\mathscr{B}$-free sets and dynamics

Bartnicka¹,

Kasjan²,

Kułaga-Przymus³

et al. 2015

Preprint

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Let B ⊂ N and let η ∈ {0, 1} Z be the characteristic function of the set F B := Z \ b∈B bZ of B-free numbers. Consider the subshift (S, X η ), where X η is the closure of the orbit of η under the left shift S. In case when B = {p 2 : p is prime} the dynamics of (S, X η ) was studied by Sarnak in 2010. This special case and some generalizations, including the case ( * ) of B infinite, pairwise coprime with

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“…Then i 0 ≤ a; if not, by (31), 2 a+1 | n which is impossible. But now, again by (31), b i0 2 i0 | n which contradicts (29).…”

Section: Unique Minimal Subset (Proof Of Theorem A)mentioning

confidence: 91%

Section: Motivationmentioning

confidence: 99%

See 1 more Smart Citation

$\mathscr{B}$-free sets and dynamics

Bartnicka¹,

Kasjan²,

Kułaga-Przymus³

et al. 2015

Preprint

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show abstract

“…As mentioned before, We only search on numbers below a certain upper bound M. Such an upper bound also helps restricting the factorization of numbers to check. In [Ian05], Iannucci gave an algorithm for the smallest abundant number not divisible by the first k primes, and also its asymptotic behavior. For our need, we only need the following results.…”

Section: Search Algorithmmentioning

confidence: 99%

Searching on the boundary of abundance for odd weird numbers

Fang

2022

Preprint

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Weird numbers are abundant numbers that are not pseudoperfect. Since their introduction, the existence of odd weird numbers has been an open problem. In this work, we describe our computational effort to search for odd weird numbers, which shows their non-existence up to 10 21 . We also searched up to 10 28 for numbers with an abundance below 10 14 , to no avail. Our approach to speed up the search can be viewed as an application of reverse search in the domain of combinatorial optimization, and may be useful for other similar quest for natural numbers with special properties that depend crucially on their factorization.

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“…The basic motivating example there was the set of abundant numbers (n P Z is abundant if |n| is smaller than the sum of its (positive) proper divisors, i.e. |n| ă σp|n|q), see also more recent [93,96,108] on that subject. Also many natural questions on general B-free sets emerged.…”

Section: Introductionmentioning

confidence: 99%

Sarnak’s Conjecture: What’s New

Ferenczi

Kułaga-Przymus

Lemańczyk

2018

Lecture Notes in Mathematics

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An overview of last seven years results concerning Sarnak's conjecture on Möbius disjointness is presented, focusing on ergodic theory aspects of the conjecture.1 Most often, however not always, T will be a homeomorphism. 2 µ stands for the arithmetic Möbius function, see next sections for explanations of notions that appear in Introduction.3 To be compared with Möbius Randomness Law by Iwaniec and Kowalski [95], page 338, that any "reasonable" sequence of complex numbers is orthogonal to µ.

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On the smallest abundant number not divisible by the first $k$ primes

Cited by 4 publications

References 1 publication

$\mathscr{B}$-free sets and dynamics

$\mathscr{B}$-free sets and dynamics

Searching on the boundary of abundance for odd weird numbers

Sarnak’s Conjecture: What’s New

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