Journal of Integer Sequences, Vol. 23 (2020), Article 20.7.2

On 3- and 9-Regular Cubic Partitions


D. S. Gireesh and C. Shivashankar
Department of Mathematics
M. S. Ramaiah University of Applied Sciences
Bengaluru-560 058
Karnataka
India

M. S. Mahadeva Naika
Department of Mathematics
Bengaluru Central University
Central College Campus
Bengaluru-560 001
Karnataka
India

Abstract:

Let a3(n) and a9(n) be 3- and 9-regular cubic partitions of n. In this paper, we establish several infinite families of congruences modulo powers of 3. For example, for all non-negative integers n and $\alpha$ we find that

\begin{displaymath}a_3\left (3^{2\alpha}n+\frac{3^{2\alpha}-1}{4}\right )\equiv 0 \pmod{3^{\alpha}}\end{displaymath}

and

\begin{displaymath}a_9\left (3^{\alpha+1}n+3^{\alpha+1}-1\right )\equiv 0 \pmod{3^{\alpha+1}}.\end{displaymath}

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(Concerned with sequences A335602 A335604.)

Received November 27 2019. Revised versions received June 4 2020; June 17 2020; June 18 2020. Published in Journal of Integer Sequences, June 24 2020.

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