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Infinite Sets of ***b*-Additive and *b*-Multiplicative
Ramanujan-Hardy Numbers

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Viorel Niţică

Department of Mathematics

West Chester University of Pennsylvania

West Chester, PA 19383

USA

**Abstract:**

Let *b* a numeration base. A *b*-additive Ramanujan-Hardy
number *N* is an integer for which there exists at least one integer
*M*,
called the 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 exist infinitely many *b*-additive
Ramanujan-Hardy numbers and infinitely many additive multipliers. A
*b*-multiplicative Ramanujan-Hardy number *N* is an integer for which
there exists at least an integer *M*, called the 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 *b* ≡ 4 (mod 6), and for *b* = 2,
there exist infinitely many
*b*-multiplicative Ramanujan-Hardy numbers and infinitely many
multiplicative multipliers. If *b* even, *b* ≡ 0 (mod 3) or
*b* ≡ 2 (mod 3), we show there exist
infinitely many numeration bases for which
there exist infinitely many *b*-multiplicative Ramanujan-Hardy numbers
and infinitely many multiplicative multipliers.
These results completely answer two questions and partially answer
two other questions asked in a previous paper of the author.

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(Concerned with sequences
A005349
A067030
A305130
A305131.)

Received December 26 2018; revised versions received March 14 2019; March 30 2019; April
4 2019.
Published in *Journal of Integer Sequences*,
May 24 2019.

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