Some results on $3$-cores
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- by Nayandeep Deka Baruah and Kallol Nath PDF
- Proc. Amer. Math. Soc. 142 (2014), 441-448 Request permission
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 \[ a_3(n)=d_{1,3}(3n+1)-d_{2,3}(3n+1),\] 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.References
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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
- Received by editor(s): March 27, 2012
- Published electronically: November 4, 2013
- Communicated by: Ken Ono
- © Copyright 2013 American Mathematical Society
- 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
- MathSciNet review: 3133986