How to change the height of the sum sign

Question

I would prefer to write the definition as in equation (2). However the sum sign is too low in this case. Is there a way to make it higher?

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation} 
\ell_{p}(n)= \sum_{k=1}^{\infty}{{\genfrac{\lfloor}{\rfloor}{}{}{n}{p^{k}}}}\bmod2 
\end{equation}
\begin{equation} 
\ell_{p}(n)= \sum_{k\geq1}{{\genfrac{\lfloor}{\rfloor}{}{}{n}{p^{k}}}}\bmod2 
\end{equation}
\end{document}

Answer 1

Don't use \genfrac for this. The command is useful in the preamble to define other commands in terms of it, not in the document.

I can offer you three proposals:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}
\ell_{p}(n)= \sum_{k\geq1}\biggl\lfloor\frac{n}{p^k}\biggr\rfloor\bmod 2
\end{equation}
\begin{equation}
\ell_{p}(n)= \sum_{k\geq1}\Bigl\lfloor\frac{n}{p^{k}}\Bigr\rfloor\bmod2
\end{equation}
\begin{equation}
\ell_{p}(n)= \sum_{k\geq1}
  \mathopen{\raisebox{-.15ex}{$\Big\lfloor$}}
  \frac{n}{p^{k}}
  \mathclose{\raisebox{-.15ex}{$\Big\rfloor$}}
  \bmod2
\end{equation}
\end{document}

I would probably choose the middle one. Changing the size of \sum is not the answer.

And surely I'd use \ell for no purpose whatsoever.

Answer 2

The OP requested "I would like it to fill the space which was kept free for the upper limit in the traditional setting."

I do not recommend this approach and concur wholeheartedly with barbara and daleif. However, being an accommodating fellow, I tried to satisfy your request. Perhaps seeing it here will convince you why it is not the best of ideas.

\documentclass{article}
\usepackage{amsmath}
\usepackage{scalerel}
\usepackage{stackengine}
\stackMath
\begin{document}
\begin{equation} 
\ell_{p}(n)= \sum_{k=1}^{\infty}{{\genfrac{\lfloor}{\rfloor}{}{}{n}{p^{k}}}}\bmod2 
\end{equation}
\begin{equation} 
\ell_{p}(n)= \stackunder[1.5pt]{\displaystyle\scalerel*{\sum}{\sum^\infty}}
  {\scriptstyle k\geq1}{{\genfrac{\lfloor}{\rfloor}{}{}{n}{p^{k}}}}\bmod2 
\end{equation}
\end{document}