Let b ≥ 2 be a numeration base. In Nițică , 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  . In , 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  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  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 ( , 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 ( , 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 .