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DIVISIBILITY OF SUMS OF PARTITION NUMBERS BY MULTIPLES OF 2 AND 3

Part of: Additive number theory; partitions Combinatorics

Published online by Cambridge University Press:  22 December 2023

NAYANDEEP DEKA BARUAH Open the ORCID record for NAYANDEEP DEKA BARUAH [Opens in a new window]
NAYANDEEP DEKA BARUAH*
Affiliation:
Department of Mathematical Sciences, Tezpur University, Napaam 784028, Assam, India
*
e-mail: [email protected]
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Abstract

We show that certain sums of partition numbers are divisible by multiples of 2 and 3. For example, if $p(n)$ denotes the number of unrestricted partitions of a positive integer n (and $p(0)=1$, $p(n)=0$ for $n<0$), then for all nonnegative integers m,

$$ \begin{align*}\sum_{k=0}^\infty p(24m+23-\omega(-2k))+\sum_{k=1}^\infty p(24m+23-\omega(2k))\equiv 0~ (\text{mod}~144),\end{align*} $$

where $\omega (k)=k(3k+1)/2$.

Keywords

partition functionsum of partition numbersℓ-regular overpartitionssingular overpartitions

MSC classification

Primary: 11P81: Elementary theory of partitions
Secondary: 05A17: Partitions of integers 11P83: Partitions; congruences and congruential restrictions 05A19: Combinatorial identities, bijective combinatorics
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 110 , Issue 2 , October 2024 , pp. 271 - 279
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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