Journal of Integer Sequences, Vol. 15 (2012), Article 12.8.1 |

Italy

**Abstract:**

Let *b* ≥ 2 be a fixed integer.
Let *s*_{b}(*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
*s*_{b}(*n*),
*s*_{b}(*n* + *q*),
..., *s*_{b}(*n* + *q*(*k*-1))
are (pairwise) distinct.
More specifically, let *L*_{b,q} denote the supremum of *k*
as *n* varies in the set of nonnegative integers **N**.
We show that *L*_{b,q} is bounded from above and hence finite.
Then it makes sense to define μ_{b,q} as the smallest
*n* ∈ **N**
such that one can take *k* = *L*_{b,q}.
We provide upper and lower bounds for μ_{b,q}.
Furthermore, we derive explicit formulas for *L*_{b,1}
and μ_{b,1}.
Lastly, we give a constructive proof that *L*_{b,q}
is unbounded with respect to *q*.

(Concerned with sequence A000120.)

Received August 5 2012;
revised version received September 23 2012.
Published in *Journal of Integer Sequences*, October 2 2012.

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