Infinite Sets of Related b-wARH Pairs

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1. Introduction

Let b ≥ 2 be a numeration base. In Nițică [1], motivated by some properties of the taxicab number, 1729, we introduced the class of b-additive Ramanujan-Hardy (or b-ARH) numbers. It consists of non-negative integers N for which there exist at least an integer M ≥ 1 such that the product of M and the sum of base b digits of N, added to the reversal of the product, give N. Many examples of b-ARH numbers can be found in [1] [2]. In [3], we introduced the class of b-weak-additive Ramanujan-Hardy (or b-wARH) numbers. It consists of non-negative integers N for which there exist at least an integer A ≥ 0, such that the sum of A and the sum of base b digits of N, added to the reversal of the sum, give N. It is shown in [3] that the class of b-wARH numbers contains the class of b-ARH numbers. Moreover, the class of b-wARH numbers contains all numerical palindromes with an even number of digits or with an odd number of digits and the middle digit even.

We say that a pair of b-wARH numbers are related of degree d ≥ 0 if their difference is d. Our main result shows, for all numeration base b ≥ 2 an infinity of degrees d for which there exists an infinity of pairs of b-wARH numbers related of degree d. Our main result leaves open the case when b = 10 and d = 2, which is of strong particular interest and for which Table 1 in [3] suggests a positive answer. This case is solved by following example.

Example 1. The palindromes ${9}^{\wedge k}$ and ${10}^{\wedge k-2}1,k\ge 1$ are a pair of 10-wARH numbers separated of degree 2.

2. The Statement of the Main Result

Let ${s}_{b}\left(N\right)$ denote the sum of base b digits of integer N. If x is a string of digits, let ${\left(x\right)}^{\wedge k}$ denote the base 10 integer obtained by repeating x k-times. Let ${\left[x\right]}_{b}$ denote the value of the string x in base b. If N is an integer, let ${N}^{\mathcal{R}}$ denote the reversal of N, that is, the number obtained from N writing its digits in reverse order. The operation of taking the reversal is dependent on the base. In the definition of a b-ARH number or a b-wARH number N we take the reversal of the base b representation of ${s}_{b}\left(N\right)\cdot M$, respectively ${s}_{b}\left(N\right)+A$. The following Theorem is our main result.

Theorem 2. For all numeration bases b ≥ 2 there exists an infinity of degrees d ≥ 0 for which there exists an infinity of pairs of b-wARH numbers related of degree d.

Theorem 2 is proved in Section 3. The following Theorem is ( [2], Theorem 1) and it is a crucial ingredient in the proof of our main result, Theorem 2.

Theorem 3. Let α ≥ 1 integer, b ≥ α + 1 integer, and $k={\left(1+\alpha \right)}^{l},l\ge 0$. Assume $b\equiv 2+\alpha \left(\mathrm{mod}2+2\alpha \right)$. Define ${N}_{k}={\left[{\left(1\alpha \right)}^{\wedge k}\right]}_{b}$. Then there exists M ≥ 0 integer such that

${s}_{b}\left({N}_{k}\right)\cdot M={\left({s}_{b}\left({N}_{k}\right)\cdot M\right)}^{R}=\frac{{N}_{k}}{2}$.

In particular, the numbers ${N}_{k},k\ge 1$, are b-ARH numbers and consequently also b-wARH numbers.

Remark 4. The particular case b = 10, α = 2, of Theorem 2, which gives ${N}_{k}={\left(12\right)}^{{3}^{l}}$, is also covered by ( [1], Example 10). Theorem 3 does not give any information if b = 2.

3. Proof of Theorem 2

Proof. If b ≥ 3 Theorem 3 can be applied to $\alpha =b-2$. This gives the b-wARH numbers ${N}_{k}={\left[{\left(1\alpha \right)}^{\wedge k}\right]}_{b}$ for $k={\left(1+\alpha \right)}^{l},l\ge 0$. Consider now the degrees ${d}_{q}={\left[1{\left({b}^{2}-4b+3\right)}^{\wedge q}\right]}_{b},q\ge 1$.

Using that ${\left[1\alpha \right]}_{b}+{\left[1\left({b}^{2}-4b+3\right)\right]}_{b}={\left[1\alpha \right]}_{b}$, the following computation, in which the right hand side is a palindrome with an even number of digits, shows that the numbers ${N}_{k}$ and $\left[{\left(1\alpha \right)}^{\wedge k-q}\right]{\left[1{\left(\alpha 1\right)}^{\wedge q}\right]}_{b}$ form a pair of b-w ARH numbers separated of degree ${d}_{q}$.

${\left[{\left(1\alpha \right)}^{\wedge k}\right]}_{b}+{\left[1{\left({b}^{2}-4b+3\right)}^{\wedge q}\right]}_{b}=\left[{\left(1\alpha \right)}^{\wedge k-q}\right]{\left[1{\left(\alpha 1\right)}^{\wedge q}\right]}_{b}$

Assuming $k\ge q$, this finishes the proof of the theorem if b ≥ 3. Assume now b = 2. Consider the degrees ${d}_{k,q}={\left[{1}^{\wedge k}{0}^{\wedge q}\right]}_{2},k\ge \text{1},q\ge \text{1}$. Let S be a string of length q with 0 and 1 digits. The following computation shows that the palindromes ${\left[S{10}^{\wedge k}1{S}^{R}\right]}_{2}$ and ${\left[S{\left(1\right)}^{\wedge k+2}{S}^{R}\right]}_{2}$ form a pair of 2-wARH numbers separated of degree ${d}_{k,q}$.

${\left[S{10}^{\wedge k}1{S}^{R}\right]}_{2}+{\left[{1}^{\wedge k}{0}^{\wedge q}\right]}_{2}={\left[S{\left(1\right)}^{\wedge k+2}{S}^{R}\right]}_{2}$.

References

[1] Nițică, V. (2018) About Some Relatives of the Taxicab Number. Journal of Integer Sequences, 21, Article 18.9.4.

[2] Nițică, V. (2019) Infinite Sets of b-Additive and b-Multiplicative Ramanujan-Hardy Numbers. Journal of Integer Sequences, 22, Article 19.4.3.

[3] Nițică, V. and Török, A. About Some Relatives of Palindromes. arXiv:1908.00713.

[4] Nițică, V. High Degree b-Niven Numbers, to Appear in Integers.

http://arxiv.org/abs/1807.02573