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In the study of the $2$ -adic sum of digits function $S_{2} (n)$ , the arithmetical function $u (0) = 0$ and $u (n) = {(- 1)}^{n - 1}$ for $n \geq 1$ plays a very important role. In this paper, we firstly generalize the relation between $S_{2} (n)$ and $u (n)$ to a bijective relation between arithmetical functions. And as an application, we investigate some aspects of the sum of digits functions $S_{𝒢} (n)$ induced by binary infinite Gray codes $𝒢$ . We can show that the difference of the sum of digits function, $S_{𝒢} (n) - S_{𝒢} (n - 1)$ , is realized by an automaton. And the summation formula of the sum...

Remark on multiplicative functions.

Andrzej Makowski (1972)

Elemente der Mathematik

Remarks on arithmetical functions $a_{p} (n), γ (n), τ (n)$ .

Fehér, Zoltán, László, Béla, Maǎj, Martin, Šalát, Tibor (2006)

Annales Mathematicae et Informaticae

Remarks on generalized Ramanujan sums and even functions.

Tóth, László (2004)

Acta Mathematica Academiae Paedagogicae Nyí regyháziensis. New Series [electronic only]

Remarks on maximum and minimum exponents in factoring

Andrzej Schinzel, Tibor Šalát (1994)

Mathematica Slovaca

Remarks on number theory I. On primitive α-abundant numbers

P Erdös (1959)

Acta Arithmetica

Remarks on number theory II. Some problems on the σ function

Paul Erdös (1959)

Acta Arithmetica

Remarks on some number-theoretical functional equations.

A. Ivic, W. Scharz (1980)

Aequationes mathematicae

Remarks to the paper of the E. Cohen "Arithmetical functions associated with arbitrary sets of integers"

(1959)

Acta Arithmetica

Remarque au sujet de la fonction $ζ_{1} (n)$ qui exprime la somme des diviseurs de $n$ .

Liouville, J. (1869)

Journal de Mathématiques Pures et Appliquées

Répartition des valeurs d'une classe de fonctions multiplicatives

A. Smati (1992)

Acta Arithmetica

Représentation multiplicative des entiers à l'aide de l'ensemble P + 1

J. Meyer (1983)

Acta Arithmetica

Représentation multiplicative des entiers et fonctions multiplicatives

Jacques MEYER (1981/1982)

Seminaire de Théorie des Nombres de Bordeaux

Representation numbers of five sextenary quadratic forms

Ernest X. W. Xia, Olivia X. M. Yao, A. F. Y. Zhao (2015)

Colloquium Mathematicae

For nonnegative integers a, b, c and positive integer n, let N(a,b,c;n) denote the number of representations of n by the form $\sum_{i = 1}^{a} (x ²_{i} + x_{i} y_{i} + y ²_{i}) + 2 \sum_{j = 1}^{b} (u ²_{j} + u_{j} v_{j} + v ²_{j}) + 4 \sum_{k = 1}^{c} (r ²_{k} + r_{k} s_{k} + s ²_{k})$ . Explicit formulas for N(a,b,c;n) for some small values were determined by Alaca, Alaca and Williams, by Chan and Cooper, by Köklüce, and by Lomadze. We establish formulas for N(2,1,0;n), N(2,0,1;n), N(1,2,0;n), N(1,0,2;n) and N(1,1,1;n) by employing the (p, k)-parametrization of three 2-dimensional theta functions due to Alaca, Alaca and Williams.

Representation numbers of stars.

Akhtar, Reza, Evans, Anthony B., Pritikin, Dan (2010)

Integers

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