• Corpus ID: 119319305

Reciprocal Sum of Palindromes

@article{Phunphayap2019ReciprocalSO,
  title={Reciprocal Sum of Palindromes},
  author={Phakhinkon Napp Phunphayap and Prapanpong Pongsriiam},
  journal={J. Integer Seq.},
  year={2019},
  volume={22},
  pages={19.8.6}
}
A positive integer $n$ is said to be a palindrome in base $b$ (or $b$-adic palindrome) if the representation of $n = (a_k a_{k-1} \cdots a_0)_b$ in base $b$ with $a_k \neq 0$ has the symmetric property $a_{k-i} = a_i$ for every $i=0,1,2,\ldots ,k$. Let $s_b$ be the reciprocal sum of all $b$-adic palindromes. It is not difficult to show that $s_b$ converges. In this article, we obtain upper and lower bounds for $s_b$ and the inequality $s_{b} <s_{b'}$ for $2\leq b<b'$. Its consequences and some… 
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References

SHOWING 1-10 OF 40 REFERENCES
Exact Formulas for the Number of Palindromes up to a Given Positive Integer
Let n ≥ 0 and b ≥ 2 be integers. Then n is said to be a palindrome in base b (or b-adic palindrome) if n = 0 or n ≥ 1 and the representation of n = (akak−1 · · · a1a0)b in base b with ak 6= 0 has the
Sums of Palindromes: an Approach via Automata
TLDR
It is proved, using a decision procedure based on automata, that every natural number is the sum of at most 4 natural numbers whose base-2 representation is a palindrome, here the constant 4 is optimal.
On The Palindromic Complexity Of Infinite Words
TLDR
It is proved that there exist periodic words having the maximum number of palindromes as in the case of Sturmian words, by providing a simple and easy to check condition.
Log‐Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometry a
A sequence a,, a,, . . . , a, of real numbers is said to be unimodal if for some 0 s j _c n we have a, 5 a , 5 . . 5 ai 2 a,,, 2 . . 2 a,, and is said to be logarithmically concave (or log-concave
Every positive integer is a sum of three palindromes
TLDR
It is proved that any positive integer can be written as a sum of three palindromes in base g as a result of the inequality of the following type.
Approximation to real numbers by cubic algebraic integers I
In 1969, H. Davenport and W. M. Schmidt studied the problem of approximation to a real number $\xi$ by algebraic integers of degree at most 3. They did so, using geometry of numbers, by resorting to
The On-Line Encyclopedia of Integer Sequences
  • N. Sloane
  • Mathematics, Computer Science
    Electron. J. Comb.
  • 1994
TLDR
The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used.
Palindromes in Different Bases: A Conjecture of J. Ernest Wilkins
Abstract We show that there exist exactly 203 positive integers N such that for some integer d ≥ 2 this number is a d-digit palindrome base 10 as well as a d-digit palindrome for some base b
Prime divisors of palindromes
TLDR
It is shown that for any sufficiently large value of $L$ there exists a palindrome $n\in{\mathcal P}_L$ with a prime factor of size at least $(\log n)^{2+o(1)}$.
On the least number of palindromes contained in an infinite word
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2
3
4
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