OFFSET
1,1
COMMENTS
The starting number itself is not counted in the trajectory, otherwise prime numbers like 113 or 227 wouldn't appear in this sequence.
The "odd Collatz trajectory" of a number k is the subset of odd numbers of the full Collatz trajectory of k.
EXAMPLE
113 is in this sequence because 113*3+1 = 340; 340/2 = 170; 170/2 = 85; 85*3+1 = 256, which goes to 1. The trajectory has 2 (> 1) tripling steps and 85 isn't a prime.
114 is not in this sequence because 114/2 = 57; 57*3+1 = 172; 172/2 = 86; 86/2 = 43, which is a prime, and this trajectory has more than 1 tripling step.
MATHEMATICA
Select[Range[3, 60000], And[Count[#, _?OddQ] > 1, NoneTrue[Rest@ #, PrimeQ]] &@ NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, #, # > 2 &, 1, Infinity, -1] &] (* Michael De Vlieger, Nov 07 2018 *)
PROG
(Java)
for(int i = 0; i < DIM; i++) {
if(!collatzAtLeastOnePrime(c) && collatzTriplingSteps(c) > 1)
System.out.print(c + ", ");
}
boolean collatzAtLeastOnePrime(int i) {
//first step outside the while loop...
if(i % 2 == 0)
i /= 2;
else
i = 3 * i + 1;
//...otherwise prime numbers like 113 or 227 would be excluded
while(i > 1) {
if(i % 2 == 0) {
i /= 2;
}
else {
if(BigInteger.valueOf(i).isProbablePrime(10))
return true;
i = 3 * i + 1;
}
}
return false;
}
CROSSREFS
KEYWORD
nonn
AUTHOR
Alessandro Polcini, Oct 10 2018
STATUS
approved