There's two features in this picture that interest me:

https://grantjenks.com/wiki/_detail/projects/automata/primediffabs500x1.jpg%3Fid=projects:automata:absolute_differences_in_consecutive_prime_numbers.html

One is at coordinate (165, 363), the other is at (318, 477), which may essentially be related to the Heegner number 163 (& 2*163).

You can draw a 45 degree angle and find another feature of interest at (250, 450) -- that's my estimate based on the features at (240, 422) & (254, 437):

This makes me interested in prime(54), and prime(38) / prime(70).

I'm leaning towards 163, 251, 342 now.

Now I'm leaning towards 163, 251, 339:

This image shows a line that essentially is supposed to represent 339 at location (338, 367):

For the above image, you can't see the bottom right of the line segment, but it seems to have 5/20 pixels to the left, unlike the other one which is a 50:50 split.

I'm a little interested in:

prime(20)*e ~= 193
prime(30)*pi ~= 355 = 5*prime(20)

(339+71)/(5/2) = 163+1 = 41*4

So if sigma(n) < 2*n - 1 (powers of 2), I think n cannot be a practical number. I'm curious if there is an upper limit where n must be a practical number.

Creating code isn't that hard, although sigma and practical numbers are obscure-ish.

from itertools import chain, cycle, accumulate, combinations
from typing import List, Tuple

# %% Factors

def factors5(n: int) -> List[int]:
"""Factors of n, (but not n)."""
def prime_powers(n):
# c goes through 2, 3, 5, then the infinite (6n+1, 6n+5) series
for c in accumulate(chain([2, 1, 2], cycle([2,4]))):
if c*c > n: break
if n%c: continue
d,p = (), c
...

...

while not n%c:
n,p,d = n//c, p*c, d + (p,)
yield(d)
if n > 1: yield((n,))

r = [1]
for e in prime_powers(n):
r += [a*b for a in r for b in e]
return r[:-1]

# %% Powerset

def powerset(s: List[int]) -> List[Tuple[int, ...]]:
"""powerset([1,2,3]) --> () (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3) ."""
return chain.from_iterable(combinations(s, r) for r in range(1, len(s)+1))
...

...

# %% Practical number

def is_practical(x: int) -> bool:
"""Practical number test with factor reverse sort and short-circuiting."""

if x == 1:
return True
if x % 2:
return False # No Odd number more than 1
mult_4_or_6 = (x % 4 == 0) or (x % 6 == 0)
if x > 2 and not mult_4_or_6:
return False # If > 2 then must be a divisor of 4 or 6
...

...

f = sorted(factors5(x), reverse=True)
if sum(f) < x - 1:
return False # Never get x-1
ps = powerset(f)

found = set()
for nps in ps:
if len(found) < x - 1:
y = sum(nps)
if 1 <= y < x:
found.add(y)
else:
break # Short-circuiting the loop.

return len(found) == x - 1
...

...

if __name__ == '__main__':
highest_practical = 1
lowest_practical = 1
highest_nonpractical = 1
lowest_nonpractical = 1

highest_practical_value = 0
lowest_practical_value = 1e10
highest_nonpractical_value = 0
lowest_nonpractical_value = 1e10
...

...

max = 3000
for x in range(1, max + 1):
normalized_sigma_value = sigma(x, 1)/x
if is_practical(x):
if normalized_sigma_value < lowest_practical_value:
lowest_practical = x
lowest_practical_value = normalized_sigma_value
...

...
elif normalized_sigma_value > highest_practical_value:
highest_practical = x
highest_practical_value = normalized_sigma_value
else:
normalized_sigma_value = sigma(x, 1)/x
if normalized_sigma_value < lowest_nonpractical_value:
lowest_nonpractical = x
lowest_nonpractical_value = normalized_sigma_value
...

...
elif normalized_sigma_value > highest_nonpractical_value:
highest_nonpractical = x
highest_nonpractical_value = normalized_sigma_value

print(lowest_practical, " (", float(lowest_practical_value), ") - ", highest_practical, " (", float(highest_practical_value), ")")
print(lowest_nonpractical, " (", float(lowest_nonpractical_value), ") - ", highest_nonpractical, " (", float(highest_nonpractical_value), ")")

Oh, right, the output is:

1 ( 1.0 ) - 2520 ( 3.7142857142857144 )
2999 ( 1.0003334444814937 ) - 1608 ( 2.537313432835821 )

Basically, the lower values (both near 1) are uninteresting.

The upper values seem to increase without bound, but maybe the ratio between them is predictable. highest_nonpractical_value may have a strict bound, but I need to get a more efficient algorithm set to figure that out.

I forgot to paste some modifications from yesterday (mostly after midnight):

Looking at records over time. I think I'm interested in the narrow band starting at the perfect numbers and going up to a factor of 2 of the upper limit: https://en.wikipedia.org/wiki/Divisor_function#Growth_rate