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DIVISIBILITY OF SUMS OF PARTITION NUMBERS BY MULTIPLES OF 2 AND 3
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 110 / Issue 2 / October 2024
- Published online by Cambridge University Press:
- 22 December 2023, pp. 271-279
- Print publication:
- October 2024
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- Article
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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
$\omega (k)=k(3k+1)/2$.
