Is there an expression for $(S/k)$ where $S=\sum_{n=1}^\infty n$ and $k \in \mathbb{Z}$?

Question

Given that $S=\sum_{n=1}^\infty n=-1/12$ (for an explanation see this question or this video from Youtube)

For example if $k=4$:

$(S/4)=1/4+2/4+3/4+1+5/4+6/4+7/4+2+9/4...$

Please edit to improve or if necessary!

Answer 1

This has been hashed and rehashed ad infinitum (!) but, obviously,
$$ S=-\frac1{12}\implies\frac14S=-\frac1{48}. $$ And at the same time, since $S$ is $$ S=\sum_{n=1}^\infty\frac1n, $$ then, "obviously", $$ S=\sum_{n\ \text{even}}\frac1n+\sum_{n\ \text{odd}}\frac1n\geqslant\sum_{n\ \text{even}}\frac1n=\sum_{n=1}^\infty\frac1{2n}=\frac12\sum_{n=1}^\infty\frac1n=\frac12S, $$ that is, $$ -\frac1{12}\geqslant\frac12\left(-\frac1{12}\right)=-\frac1{24}, $$ which opens up some fascinating possibilities, such as, sooner or later, $$ -1\geqslant0. $$