A175871 a(0) = 2; a(n) = a(n - 1) * 3 + 1 if a(n - 1) is prime, or a(n - 1) / (smallest prime factor) if it is composite.

2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8

Offset

0,1

Comments

a(n) repeats itself after 16 iterations. Peak is a(6) = 52.

The function is similar in nature to Collatz's 3x+1 problem, except that it deals with primality instead of parity.

Links

Table of n, a(n) for n=0..78.

Wikipedia, Divisor

Wikipedia, Prime number

Example

a(0) = 2
a(1) = 2 * 3 + 1 = 7, because a(0) was prime.
a(2) = 7 * 3 + 1 = 22, because a(1) was prime.
a(3) = 22 / 2 = 11, because the smallest prime factor of a(2) was 2.

Mathematica

NestList[If[PrimeQ[#], 3#+1, #/FactorInteger[#][[1, 1]]]&, 2, 80] (* or  *) PadRight[{}, 80, {2, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4}] (* Harvey P. Dale, Aug 05 2023 *)

Prog

(Python)
import math
from sympy import isprime
a = [2]
while not a[ -1] in a[:-1]:
    if isprime(a[ -1]):
        a.append(a[ -1] * 3 + 1)
    else:
        for div in range(2, int(math.sqrt(a[ -1])) + 1):
            if not a[ -1] % div:
                a.append(a[ -1] // div)
                break
print(a)

Crossrefs

Cf. A000040, A175867.

Sequence in context: A076716 A088591 A229493 * A137107 A284921 A174236

Adjacent sequences: A175868 A175869 A175870 * A175872 A175873 A175874

Keyword

easy,nonn

Author

Grant Garcia, Oct 02 2010

Status

approved