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Some results on $ 3$-cores


Authors: Nayandeep Deka Baruah and Kallol Nath
Journal: Proc. Amer. Math. Soc. 142 (2014), 441-448
MSC (2010): Primary 11P83; Secondary 05A17
DOI: https://doi.org/10.1090/S0002-9939-2013-11784-3
Published electronically: November 4, 2013
MathSciNet review: 3133986
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Abstract: We prove that if $ u(n)$ denotes the number of representations of a nonnegative integer $ n$ in the form $ x^2+3y^2$ with $ x,y\in \mathbb{Z}$, and $ a_3(n)$ is the number of $ 3$-cores of $ n$, then $ u(12n+4)=6a_3(n)$. With the help of a classical result by L. Lorenz in 1871, we also deduce that

$\displaystyle a_3(n)=d_{1,3}(3n+1)-d_{2,3}(3n+1),$

where $ d_{r,3}(n)$ is the number of divisors of $ n$ congruent to $ r$ (mod $ 3$), a result proved earlier by Granville and Ono by using the theory of modular forms and by Hirschhorn and Sellers with the help of elementary generating function manipulations.

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Additional Information

Nayandeep Deka Baruah
Affiliation: Department of Mathematical Sciences, Tezpur University, Sonitpur, PIN-784028, India
Email: nayan@tezu.ernet.in

Kallol Nath
Affiliation: Department of Mathematical Sciences, Tezpur University, Sonitpur, PIN-784028, India
Email: kallol08@tezu.ernet.in

DOI: https://doi.org/10.1090/S0002-9939-2013-11784-3
Keywords: Partitions, $t$-cores, theta functions, dissection
Received by editor(s): March 27, 2012
Published electronically: November 4, 2013
Communicated by: Ken Ono
Article copyright: © Copyright 2013 American Mathematical Society

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