# On the counting function for the generalized Niven numbers

Ryan Daileda[1]; Jessica Jou[2]; Robert Lemke-Oliver[3]; Elizabeth Rossolimo[4]; Enrique Treviño[5]

• [1] Mathematics Department Trinity University One Trinity Place San Antonio, TX 78212-7200, USA
• [2] 678-2 Azumi, Ichinomiya-cho Shiso-shi, Hyogo 671-4131 Japan
• [3] Deparment of Mathematics University of Wisconsin Madison 480 Lincoln Dr Madison, WI 53706 USA
• [4] Department of Mathematics and Statistics Lederle Graduate Research Tower Box 34515 University of Massachusetts Amherst Amherst, MA 01003-9305, USA
• [5] Department of Mathematics 6188 Kemeny Hall Dartmouth College Hanover, NH 03755-3551, USA
• Volume: 21, Issue: 3, page 503-515
• ISSN: 1246-7405

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## Abstract

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Given an integer base $q\ge 2$ and a completely $q$-additive arithmetic function $f$ taking integer values, we deduce an asymptotic expression for the counting function${N}_{f}\left(x\right)=#\left\{0\le n<x\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}f\left(n\right)\mid n\right\}$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.

## How to cite

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Daileda, Ryan, et al. "On the counting function for the generalized Niven numbers." Journal de Théorie des Nombres de Bordeaux 21.3 (2009): 503-515. <http://eudml.org/doc/10895>.

@article{Daileda2009,
abstract = {Given an integer base $q \ge 2$ and a completely $q$-additive arithmetic function $f$ taking integer values, we deduce an asymptotic expression for the counting function\begin\{equation*\} N\_f(x) = \# \left\lbrace 0 \le n &lt; x \, | \, f(n) \mid n \right\rbrace \end\{equation*\}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.},
affiliation = {Mathematics Department Trinity University One Trinity Place San Antonio, TX 78212-7200, USA; 678-2 Azumi, Ichinomiya-cho Shiso-shi, Hyogo 671-4131 Japan; Deparment of Mathematics University of Wisconsin Madison 480 Lincoln Dr Madison, WI 53706 USA; Department of Mathematics and Statistics Lederle Graduate Research Tower Box 34515 University of Massachusetts Amherst Amherst, MA 01003-9305, USA; Department of Mathematics 6188 Kemeny Hall Dartmouth College Hanover, NH 03755-3551, USA},
author = {Daileda, Ryan, Jou, Jessica, Lemke-Oliver, Robert, Rossolimo, Elizabeth, Treviño, Enrique},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Niven number; Harshad number; sum of digits function; completely -additive arithmetic function; asymptotic formula},
language = {eng},
number = {3},
pages = {503-515},
publisher = {Université Bordeaux 1},
title = {On the counting function for the generalized Niven numbers},
url = {http://eudml.org/doc/10895},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Daileda, Ryan
AU - Jou, Jessica
AU - Lemke-Oliver, Robert
AU - Rossolimo, Elizabeth
AU - Treviño, Enrique
TI - On the counting function for the generalized Niven numbers
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 3
SP - 503
EP - 515
AB - Given an integer base $q \ge 2$ and a completely $q$-additive arithmetic function $f$ taking integer values, we deduce an asymptotic expression for the counting function\begin{equation*} N_f(x) = \# \left\lbrace 0 \le n &lt; x \, | \, f(n) \mid n \right\rbrace \end{equation*}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.
LA - eng
KW - Niven number; Harshad number; sum of digits function; completely -additive arithmetic function; asymptotic formula
UR - http://eudml.org/doc/10895
ER -

## References

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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. Zbl0582.10007MR809074
3. C. N. Cooper, R. E. Kennedy, A partial asymptotic formula for the Niven numbers. Fibonacci Quart. 26 (1988), 163–168. Zbl0644.10009MR938592
4. C. N. Cooper, R. E. Kennedy, Chebyshev’s inequality and natural density. Amer. Math. Monthly 96 (1989), 118–124. Zbl0694.10004MR992072
5. J.-M. De Koninck, N. Doyon, On the number of Niven numbers up to $x$. Fibonacci Quart. 41 (5) (2003), 431–440. Zbl1057.11005MR2053095
6. J.-M. De Koninck, N. Doyon, I. Kátai, On the counting function for the Niven numbers. Acta Arith. 106 (3) (2003), 265–275. Zbl1023.11003MR1957109
7. H. Delange, Sur les fonctions $q$-additives ou $q$-multiplicatives. Acta Arith. 21 (1972), 285–298. Zbl0219.10062MR309891
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. Zbl1155.11345MR2166377
9. V. V. Petrov, Sums of Independent Random Variables. Ergeb. Math. Grenzgeb. 82, Springer, 1975. Zbl0322.60042MR388499

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