Network graph of n→n+φ(n), investigating different starting values from http://oeis.org/A074693, as suggested by tox#6692

phi_plus_n_graph.py

from sympy.ntheory import factorint
from sympy.ntheory.factor_ import totient
import functools
import itertools
import networkx as nx
from collections import deque
# pygraphviz is optional; you can use command-line graphviz instead if you want
import pygraphviz as pgv


@functools.lru_cache(maxsize=None)
def f(n):
    return n+totient(n)


G = nx.DiGraph()

placed = set()

# We extend the graph in an interesting manner as a compromise between plugging in
# 1,2,3,4,5,6,... and 1,f(1),f(f(1)),f(f(f(1))),...
# It proceeds something like 1,f(1),2,f(f(1)),f(2),3,f(f(f(1))),f(f(2)),f(3),4...
# with duplicates removed. This is done using the queue (deque),
# where `0` signifies proceeding along the 1,2,3,... subsequence

# current filled max
current_max = 1
heads_queue = deque([0])
for _ in range(1000):
    if len(heads_queue) == 0:
        head = 0
    else:
        head = heads_queue.popleft()
    if head == 0:
        while current_max in placed:
            current_max += 1
        head = current_max
        heads_queue.append(0)
    if head in placed:
        continue
    res = f(head)
    if res not in placed:
        heads_queue.append(res)
    placed.add(head)
    ratio = res/head
    label = "{:1.4}".format(ratio)
    # HSV
    color = f"{0.7*ratio%1} 1 0.7"
    G.add_edge(head, res, label=label, color=color)

shapes = itertools.cycle(['ellipse', 'rectangle'])
for component in nx.weakly_connected_components(G):
    shape = next(shapes)
    for node in component:
        G.nodes[node]['shape'] = shape


def factor_string(n):
    factors = factorint(n)
    # don't feel like setting up dot2tex, so HTML superscripts will have to do for now
    return "<" + "·".join([
        f"{p}<SUP>{n}</SUP>"
        for p, n in factors.items()
    ]) + ">" or "1"


for node in G:
    G.nodes[node]['label'] = factor_string(node)


A = nx.nx_agraph.to_agraph(G)
A.write("graphviz.dot")
pgv.AGraph("graphviz.dot").draw("output.png", prog="dot")