Fractal digital sums and codes

Consider the multiplicities e p 1 (n), e p 2 (n), . . . , e p k (n) in which the primes p 1 , p 2 , . . . , p k appear in the factorization of n!. We show that these multiplicities are jointly uniformly distributed modulo (m 1 , m 2 , . . . , m k ) for any fixed integers m 1 , m 2 , . . . , m k , thus improving a result of Luca and Stȃnicȃ [F. Luca, P. Stȃnicȃ, On the prime power factorization of n!, J. Number Theory 102 (2003) 298-305].To prove the theorem, we obtain a result regarding the joint distribution of several completely q-additive functions, which seems to be of independent interest.

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“…For similar results for other primes, as well as results showing that these primes are actually exceptional, we refer to [4,5,7,8].…”

Section: Example 25 According To Theorem 24 We Havementioning

confidence: 63%

Regularity of patterns in the factorization of n!

Berend

Kolesnik

2007

Journal of Number Theory

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“…Existence of such Coquet-type formulas and single precise results were already derived for more general rarefied Thue-Morse sequences, see for example [10][11][12]. An extended overview of the investigation carried out for the rarefied Thue-Morse sequences can be found in [9].…”

Section: Introductionmentioning

confidence: 91%

“…, p − 1} are often called rarefied Thue-Morse sequences. Several asymptotic properties of the sum ∑ N−1 n=0 (−1) s (pn+j) for different values of p and j were investigated for example in [3][4][5]11,12,23,24].…”

Section: Introductionmentioning

confidence: 99%

Coquet-type formulas for the rarefied weighted Thue–Morse sequence

Hofer

2011

Discrete Mathematics

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a b s t r a c tNewman proved for the classical Thue-Morse sequence, ((−1) s(n) ) n≥0 , that c 1 N λ < ∑ N−1 n=0 (−1) s(3n) < c 2 N λ for all N ∈ N with real constants λ, c 1 , c 2 satisfying c 2 > c 1 > 0 and λ = log 3/ log 4. Coquet improved this result and deducedis a nowhere-differentiable, continuous function with period 1 and η(N) ∈ {−1, 0, 1}. In this paper we obtain for the weighted version of the Thue-Morse sequence that for the sum ∑ N−1 n=0 (−1) sγ (3n+r) a Coquet-type formula exists for every r ∈ {0, 1, 2} if and only if the sequence of weights is eventually periodic. From the specific Coquet-type formulas we derive parts of the weak Newman-type results that were recently obtained by Larcher and Zellinger.

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“…Let B(n) be the sum of digits of the positive integer n written in base 2. This function represents the numbers of ones in the binary expansion of n, or from the code theory point of view, the number of nonzero digits in the bits string representing n, i.e., the so-called 'Hamming weight' of n. It is obvious that B(n) = B(2n), but for a prime p > 2 a relation between B(n) and B(pn) is less trivial [2,[4][5][6]9].…”

Section: Introductionmentioning

confidence: 99%