Ulam spirale, envolved to

[RomanM]

Hi.
Plot code was taken from here
https://community.wolfram.com/groups/-/m/t/1543056

mgonLayer[q_, m_] := Ceiling[1/2 (1 + Sqrt[-8 + m + 8 q]/Sqrt[m])] - 1

maround[q_, m_] := q - (1 + m/2 ((mgonLayer[q, m] - 1) (mgonLayer[q, m])))
mpoint[q_, m_] := (mgonLayer[q, m] AngleVector[(2 \[Pi])/m (#1 - 1)] +
                 #2 2 Sin[\[Pi]/m] AngleVector[2 \[Pi]/m (#1 - 1) + \[Pi]/2 + \[Pi]/m]) & @@ 
                  Quiet[QuotientRemainder[maround[q, m], mgonLayer[q, m]]]

ListPlot[mpoint[#, M] & /@ (Prime[#] & /@ Range[10000]), Axes -> False, AspectRatio -> Automatic]

For classic spirale, M = 4;
Lets make a little changes in the plot code:

ListPlot[mpoint[#, M] & /@ (Prime[#*#] & /@ Range[20000]), Axes -> False, AspectRatio -> Automatic]

M = 403
Sharingan
https://www.mersenneforum.org/attach...1&d=1655050840
M = 503
Rinnegan
https://www.mersenneforum.org/attach...1&d=1655051029

[RomanM]

Hello.

ListPlot[mpoint[#, 4] & /@ ((Prime[2*#] - 2*Prime[#]) & /@
    Range[15000]), Axes -> True, AspectRatio -> Automatic]

what do you think, is it possible to have a prime related function instead of

 Prime[2*#] - 2*Prime[#]

that gives a symmetry?

[RomanM]

Originally Posted by RomanM

*** that gives a symmetry?

some symmery

ListPlot[mpoint[#, 4] & /@ ((Prime[#]/EulerPhi[#]) & /@ Range[15000]),
  Axes -> True, AspectRatio -> Automatic]

on the graph is connection Euler Phi(n) and n-prime number, and it's not impossible to fast compute the n-prime number for given (full factored) n.