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On the counting function for the generalized Niven numbers. (English) Zbl 1205.11105

For an integer \(q\geq 2\), a positive integer \(n\) is said to be a \(q\)-Niven (or \(q\)-Harshad) number if \(n\) is divisible by the sum of its digits in base \(q\), and more generally a positive integer \(n\) is said to be an \(f\)-Niven number if \(f(n)\mid n\), where \(f\) is a nonzero completely \(q\)-additive arithmetic function with integer values. The purpose of this paper is to prove an asymptotic formula for the number of \(f\)-Niven numbers not exceeding \(x\) under some mild conditions on \(f\). If \(f\) is the sum of digits function to the base \(q\), the result gives the corresponding asymptotic formulas by J.-M. De Koninck, N. Doyon and I. Kátai [Acta Arith. 106, No. 3, 265–275 (2003; Zbl 1023.11003)] and C. Mauduit, C. Pomerance and A. Sárközy [Ramanujan J. 9, No. 1–2, 45–62 (2005; Zbl 1155.11345)]. Generalization to \(f\)-Niven numbers was suggested by J.-M. De Koninck, N. Doyon and I. Kátai. The significant modifications needed in this general case are also pointed out in the paper under review.

MSC:

11N37 Asymptotic results on arithmetic functions
11N64 Other results on the distribution of values or the characterization of arithmetic functions
11A63 Radix representation; digital problems
11N56 Rate of growth of arithmetic functions
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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. · Zbl 0582.10007
[3] C. N. Cooper, R. E. Kennedy, A partial asymptotic formula for the Niven numbers. Fibonacci Quart. 26 (1988), 163-168. · Zbl 0644.10009
[4] C. N. Cooper, R. E. Kennedy, Chebyshev’s inequality and natural density. Amer. Math. Monthly 96 (1989), 118-124. · 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. · 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. · Zbl 1023.11003
[7] H. Delange, Sur les fonctions \(q\)-additives ou \(q\)-multiplicatives. Acta Arith. 21 (1972), 285-298. · 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. · Zbl 1155.11345
[9] V. V. Petrov, Sums of Independent Random Variables. Ergeb. Math. Grenzgeb. 82, Springer, 1975. · Zbl 0322.60042
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