2,447 research outputs found
Stolarsky's conjecture and the sum of digits of polynomial values
Let denote the sum of the digits in the -ary expansion of an
integer . In 1978, Stolarsky showed that He conjectured that, as for , this limit
infimum should be 0 for higher powers of . We prove and generalize this
conjecture showing that for any polynomial with and and any base , For any we
give a bound on the minimal such that the ratio . Further, we give lower bounds for the number of such that
.Comment: 13 page
The sum of digits of and
Let denote the sum of the digits in the -ary expansion of an
integer . In 2005, Melfi examined the structure of such that . We extend this study to the more general case of generic and
polynomials , and obtain, in particular, a refinement of Melfi's result.
We also give a more detailed analysis of the special case , looking
at the subsets of where for fixed .Comment: 16 page
Thue-Morse at Multiples of an Integer
Let (t_n) be the classical Thue-Morse sequence defined by t_n = s_2(n) (mod
2), where s_2 is the sum of the bits in the binary representation of n. It is
well known that for any integer k>=1 the frequency of the letter "1" in the
subsequence t_0, t_k, t_{2k}, ... is asymptotically 1/2. Here we prove that for
any k there is a n<=k+4 such that t_{kn}=1. Moreover, we show that n can be
chosen to have Hamming weight <=3. This is best in a twofold sense. First,
there are infinitely many k such that t_{kn}=1 implies that n has Hamming
weight >=3. Second, we characterize all k where the minimal n equals k, k+1,
k+2, k+3, or k+4. Finally, we present some results and conjectures for the
generalized problem, where s_2 is replaced by s_b for an arbitrary base b>=2.Comment: 14 page
On a conjecture of Dekking : The sum of digits of even numbers
Let and denote by the sum-of-digits function in base . For
consider # \{0 \le n < N : \;\;s_q(2n) \equiv j \pmod q \}.
In 1983, F. M. Dekking conjectured that this quantity is greater than
and, respectively, less than for infinitely many , thereby claiming an
absence of a drift (or Newman) phenomenon. In this paper we prove his
conjecture.Comment: 6 pages, accepted by JTN
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