Otdfbt

I need this to work with any n I decide to set in TikZ (because the diagram at this point is undetermined). I don't mind setting margin manually, but no arcs except the one from node 3 to 4 connect correctly, and I'm clearly not good at working with polar coordinates.

This is a modified version of http://www.texample.net/tikz/examples/cycle/, except I would like for it to start on the left, and it needs to be clockwise.

documentclassarticle
usepackagetikz
begindocument

begincenter
begintikzpicture[scale=0.7]

def n 5
def radius 3cm
def margin 8.2 % margin in angles, depends on the radius
% https://tex.stackexchange.com/questions/47172/what-does-the-colon-in-tikz-coordinates-mean
% http://www.texample.net/tikz/examples/cycle/

foreach s in 1,...,n

node[draw, circle] at (180 - 360/n * (s - 1):radius) $s$;
draw[->, >=latex] (180 - 360/n * (s - 1)-margin:radius) 
% https://tex.stackexchange.com/questions/175016/how-is-arc-defined-in-tikz
 arc (2*margin:-(180/n) + 2* margin :(radius * 1.5);

endtikzpicture
endcenter


enddocument

Output:

documentclassarticle
usepackagetikz
begindocument
begintikzpicture

def n 5
def radius 3cm
def margin 8 % margin in angles, depends on the radius
foreach s in 1,...,n 
 node[draw, circle] at (360/n*(1-s):-radius) s;
 draw[-latex] (360/n*(1-s)-margin:-radius) 
 arc (360/n*(1-s)-margin:-360/n*s+margin:-radius);

endtikzpicture
enddocument

Following the example you linked, you can change the degree at which nodes are drawn and the direction of the arrowhead in draw:

documentclassarticle
usepackagetikz
begindocument
begintikzpicture

def n 5
def radius 3cm
def margin 8 % margin in angles, depends on the radius

foreach s in 1,...,n

 node[draw, circle] at (360/n * (n - s - 1):radius) $s$;
 draw[<-, =>latex] (360/n * (s - 1)+margin:radius) 
 arc (360/n * (s - 1)+margin:360/n * (s)-margin:radius);

endtikzpicture

As pointed out here, this is a tree, and such trees understand clockwise.

documentclass[tikz,border=3.14mm]standalone
usetikzlibrarytrees
begindocument
begintikzpicture[circ/.style=circle,draw,
level 1/.style=sibling angle=72,level distance=pi*1cm,
edge from parent/.style= draw=none]
coordinate (Center) [clockwise from=270]
 child foreach X in 1,...,5 node[circ] (X) X;
foreach X [remember=X as LastX (initially 5)] in 1,...,5
draw[-latex] (LastX) to[bend left=28] (X); 
endtikzpicture
enddocument