On the counting function for the generalized Niven numbers

Daileda, Ryan; Jou, Jessica; Lemke-Oliver, Robert; Rossolimo, Elizabeth; Treviño, Enrique

Daileda, Ryan; Jou, Jessica; Lemke-Oliver, Robert; Rossolimo, Elizabeth; Treviño, Enrique

Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, p. 503-515

Abstract
Résumé

Given an integer base $q \geq 2$ and a completely $q$ -additive arithmetic function $f$ taking integer values, we deduce an asymptotic expression for the counting function

N_{f} (x) = # \{0 \leq n < x | f (n) ∣ n\}

under a mild restriction on the values of $f$ . When $f = s_{q}$ , the base $q$ sum of digits function, the integers counted by $N_{f}$ are the so-called base $q$ Niven numbers, and our result provides a generalization of the asymptotic known in that case.

Étant donnés $q \geq 2$ un entier naturel et $f$ une fonction complètement q-additive à valeurs dans l’ensemble des nombres entiers relatifs, on calcule une expression asymptotique de la fonction $N_{f}$ qui à $x$ associe la cardinalité de l’ensemble

{0 \leq n < x | f (n) ∣ n}

quand les valeurs de $f$ sont soumises à une petite restriction. Dans le cas où $f = s_{q}$ , la somme des chiffres d’un nombre en base $q$ , les valeurs de la function $N_{f}$ comptent les nombres q-Harshad. Donc, notre résultat généralise la formule asymptotique dans ce cas.

MR 2605530 | Zbl 1205.11105

DOI : https://doi.org/10.5802/jtnb.685
Classification: 11A25, 11A63, 11K65

@article{JTNB_2009__21_3_503_0,
     author = {Daileda, Ryan and Jou, Jessica and Lemke-Oliver, Robert and Rossolimo, Elizabeth and Trevi\~no, Enrique},
     title = {On the counting function for the generalized Niven numbers},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {3},
     year = {2009},
     pages = {503-515},
     doi = {10.5802/jtnb.685},
     mrnumber = {2605530},
     zbl = {1205.11105},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2009__21_3_503_0}
}

Daileda, Ryan; Jou, Jessica; Lemke-Oliver, Robert; Rossolimo, Elizabeth; Treviño, Enrique. On the counting function for the generalized Niven numbers. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 503-515. doi : 10.5802/jtnb.685. http://www.numdam.org/item/JTNB_2009__21_3_503_0/

References

[1] C. N. Cooper, R. E. Kennedy, On the natural density of the Niven numbers. College Math. J. 15 (1984), 309–312.

[2] C. N. Cooper, R. E. Kennedy, On an asymptotic formula for the Niven numbers. Internat. J. Math. Sci. 8 (1985), 537–543. | MR 809074 | Zbl 0582.10007

[3] C. N. Cooper, R. E. Kennedy, A partial asymptotic formula for the Niven numbers. Fibonacci Quart. 26 (1988), 163–168. | MR 938592 | Zbl 0644.10009

[4] C. N. Cooper, R. E. Kennedy, Chebyshev’s inequality and natural density. Amer. Math. Monthly 96 (1989), 118–124. | MR 992072 | Zbl 0694.10004

[5] J.-M. De Koninck, N. Doyon, On the number of Niven numbers up to $x$ . Fibonacci Quart. 41 (5) (2003), 431–440. | MR 2053095 | Zbl 1057.11005

[6] J.-M. De Koninck, N. Doyon, I. Kátai, On the counting function for the Niven numbers. Acta Arith. 106 (3) (2003), 265–275. | MR 1957109 | Zbl 1023.11003

[7] H. Delange, Sur les fonctions $q$ -additives ou $q$ -multiplicatives. Acta Arith. 21 (1972), 285–298. | MR 309891 | Zbl 0219.10062

[8] C. Mauduit, C. Pomerance, A. Sárközy, On the distribution in residue classes of integers with a fixed sum of digits. Ramanujan J. 9 (1-2) (2005), 45–62. | MR 2166377 | Zbl 1155.11345

[9] V. V. Petrov, Sums of Independent Random Variables. Ergeb. Math. Grenzgeb. 82, Springer, 1975. | MR 388499 | Zbl 0322.60042