PreprintCC BY 4.0arXiv
Divisibility Properties of Odd Part and Even Part Sums in Cubic Partitions
Abstract
A cubic partition of a positive integer n is a partition of n in which even parts may appear in two distinct colors. The purpose of this paper is to study the arithmetic functions COS(n) and CES(n), which denote the sums of all odd and even parts, respectively, in all cubic partitions of n. We obtain their generating functions and divisor-sum formulas, and establish Ramanujan-type congruences modulo $3$, $4$, $5$, and $7$. To prove these congruences, we use Lambert series, classical theta-function identities, dissections, coefficient extraction, and weighted Euler identities.
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BibTeX
@misc{bardhan2026divisibility,
title = {{Divisibility Properties of Odd Part and Even Part Sums in Cubic Partitions}},
author = {Gaurab Bardhan and Nipen Saikia},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05196},
eprint = {2611.05196},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05196},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Divisibility Properties of Odd Part and Even Part Sums in Cubic Partitions
AU - Gaurab Bardhan
AU - Nipen Saikia
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05196
UR - https://arxiv.org/abs/2611.05196
AB - A cubic partition of a positive integer n is a partition of n in which even parts may appear in two distinct colors. The purpose of this paper is to study the arithmetic functions COS(n) and CES(n), which denote the sums of all odd and even parts, respectively, in all cubic partitions of n. We obtain their generating functions and divisor-sum formulas, and establish Ramanujan-type congruences modulo $3$, $4$, $5$, and $7$. To prove these congruences, we use Lambert series, classical theta-function identities, dissections, coefficient extraction, and weighted Euler identities.
ER -