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.
Full version: pdf,
dvi,
ps,
latex
(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.
Return to
Journal of Integer Sequences home page