Divisibility and distribution of $MEX$ related integer partitions of Andrews and Newman
In: International Journal of Number Theory, Vol. 19 (2023); (2023)
Online
report
Andrews and Newman introduced the minimal excludant or ``$mex$'' function for an integer partition $\pi$ of a positive integer $n$, $mex(\pi)$, as the smallest positive integer that is not a part of $\pi$. They defined $\sigma mex(n)$ to be the sum of $mex(\pi)$ taken over all partitions $\pi$ of $n$. We prove infinite families of congruence and multiplicative formulas for $\sigma mex(n)$. By restricting to the part of $\pi$, Andrews and Newman also introduced $moex(\pi)$ to be the smallest odd integer that is not a part of $\pi$ and $\sigma moex(n)$ to be the sum of $moex(\pi)$ taken over all partitions $\pi$ of $n$. In this article, we show that for any sufficiently large $X$, the number of all positive integer $n\leq X$ such that $\sigma moex(n)$ is an even (or odd) number is at least $\mathcal{O}(\log \log X)$.
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Titel: |
Divisibility and distribution of $MEX$ related integer partitions of Andrews and Newman
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Autor/in / Beteiligte Person: | Ray, Chiranjit |
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Quelle: | International Journal of Number Theory, Vol. 19 (2023); (2023) |
Veröffentlichung: | 2023 |
Medientyp: | report |
DOI: | 10.1142/S1793042123500288 |
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