Infinite sets of $b$-additive and $b$-multiplicative Ramanujan-Hardy numbers

Abstract

Let $b$ be a numeration base. A $b$-additive Ramanujan-Hardy number $N$ is an integer for which there exists at least an integer $M$, called additive multiplier, such that the product of $M$ and the sum of base $b$ digits of $N$, added to the reversal of the product, gives $N$. We show that for any $b$ there exists an infinity of $b$-additive Ramanujan-Hardy numbers and an infinity of additive multipliers. A $b$-multiplicative Ramanujan-Hardy number $N$ is an integer for which there exists at least an integer $M$, called multiplicative multiplier, such that the product of $M$ and the sum of base $b$ digits of $N$, multiplied by the reversal of the product, gives $N$. We show that for an even $b$, $b\equiv 1(\mathrm{mod}3)$, and for $b=2$, there exists an infinity of $b$-multiplicative Ramanujan-Hardy numbers and an infinity of multiplicative multipliers. These results completely answer two questions and partially answer two other questions among those asked in V. Niţică, \emph{About some relatives of the taxicab number}, arXiv:1805.10739v4.