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%I #9 Aug 05 2023 12:01:28
%S 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,
%T 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,
%U 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
%N 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.
%C a(n) repeats itself after 16 iterations. Peak is a(6) = 52.
%C The function is similar in nature to Collatz's 3x+1 problem, except that it deals with primality instead of parity.
%H Wikipedia, <a href="http://en.wikipedia.org/en/wiki/Divisor">Divisor</a>
%H Wikipedia, <a href="http://en.wikipedia.org/en/wiki/Prime_number">Prime number</a>
%e a(0) = 2
%e a(1) = 2 * 3 + 1 = 7, because a(0) was prime.
%e a(2) = 7 * 3 + 1 = 22, because a(1) was prime.
%e a(3) = 22 / 2 = 11, because the smallest prime factor of a(2) was 2.
%t 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 *)
%o (Python)
%o import math
%o from sympy import isprime
%o a = [2]
%o while not a[ -1] in a[:-1]:
%o if isprime(a[ -1]):
%o a.append(a[ -1] * 3 + 1)
%o else:
%o for div in range(2, int(math.sqrt(a[ -1])) + 1):
%o if not a[ -1] % div:
%o a.append(a[ -1] // div)
%o break
%o print(a)
%Y Cf. A000040, A175867.
%K easy,nonn
%O 0,1
%A _Grant Garcia_, Oct 02 2010
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