On Arithmetic Progressions of Integers with a Distinct Sum of Digits
Let b ≥ 2 be a fixed integer.
Let sb(n) denote the sum of digits of
the nonnegative integer n in the base-b representation.
Further let q be a positive integer.
In this paper we study the length k of arithmetic progressions
n, n + q, ..., n + q(k-1) such that
sb(n + q),
..., sb(n + q(k-1))
are (pairwise) distinct.
More specifically, let Lb,q denote the supremum of k
as n varies in the set of nonnegative integers N.
We show that Lb,q is bounded from above and hence finite.
Then it makes sense to define μb,q as the smallest
n ∈ N
such that one can take k = Lb,q.
We provide upper and lower bounds for μb,q.
Furthermore, we derive explicit formulas for Lb,1
Lastly, we give a constructive proof that Lb,q
is unbounded with respect to q.
Full version: pdf,
(Concerned with sequence
Received August 5 2012;
revised version received September 23 2012.
Published in Journal of Integer Sequences, October 2 2012.
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