Journal of Integer Sequences, Vol. 22 (2019), Article 19.4.3

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


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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