A327439 a(0)=1. If a(n-1) and n are relatively prime and a(n-1)!=1, a(n) = a(n-1) - 1. Otherwise (i.e., if a(n-1) and n share a common factor or a(n-1)=1), a(n) = a(n-1) + gcd(a(n-1),n) + 1.

1, 3, 2, 1, 3, 2, 5, 4, 9, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 3, 2, 5, 4, 9, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 3, 2, 5, 4, 7, 6, 9, 8, 11, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11

Offset: 0

Author

Nathaniel J. Strout, Feb 24 2020

Keywords

• nonn
• look

Comments

Graphically at large scales this sequence is vaguely self-similar, though in certain sections it acts in a rough manner, in particular in regions surrounding apparent cusps. See the program for a graph to zoom in on these sections.

See Python program for zoomable graph.

Links

• Nathaniel J. Strout, Table of n, a(n) for n = 0..116000

Programs

Mathematica

a[0] = 1; a[n_] := a[n] = If[a[n - 1] != 1 && CoprimeQ[n, a[n - 1]], a[n - 1] - 1, a[n - 1] + GCD[a[n - 1], n] + 1]; Array[a, 101, 0] (* Amiram Eldar, Feb 24 2020 *)
nxt[{n_,a_}]:={n+1,If[CoprimeQ[n+1,a]&&a!=1,a-1,a+GCD[a,n+1]+1]}; NestList[nxt,{0,1},70][[;;,2]] (* Harvey P. Dale, Jun 08 2024 *)

Python

import math
import matplotlib.pyplot as plt
num = 10000
x = []
y = []
# y is the main sequence
def sequence():
    a = 1
    y.append(a)
    for i in range(num):
        if (a != 1) and (math.gcd(a,i+1) == 1):
            a -= 1
        else:
            a += math.gcd(a,i+1)+1
        x.append(i)
        y.append(a)
    x.append(num)
sequence()
# code only regarding the plot.
plt.xlim(0,num)
plt.ylim(0,num)
plt.plot(x, y)
plt.xlabel('x - axis')
plt.ylabel('y - axis')
plt.title('Plot of Sequence')
plt.show()