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
and
are a pair of 10-wARH numbers separated of degree 2.
2. The Statement of the Main Result
Let
denote the sum of base b digits of integer N. If x is a string of digits, let
denote the base 10 integer obtained by repeating x k-times. Let
denote the value of the string x in base b. If N is an integer, let
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
, respectively
. 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
. Assume
. Define
. Then there exists M ≥ 0 integer such that
.
In particular, the numbers
, are b-ARH numbers and consequently also b-wARH numbers.
Remark 4. The particular case b = 10, α = 2, of Theorem 2, which gives
, 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
. This gives the b-wARH numbers
for
. Consider now the degrees
.
Using that
, the following computation, in which the right hand side is a palindrome with an even number of digits, shows that the numbers
and
form a pair of b-w ARH numbers separated of degree
.
Assuming
, this finishes the proof of the theorem if b ≥ 3. Assume now b = 2. Consider the degrees
. Let S be a string of length q with 0 and 1 digits. The following computation shows that the palindromes
and
form a pair of 2-wARH numbers separated of degree
.
.