A319936 Numbers with more than one Collatz tripling step whose odd Collatz trajectory does not contain primes.

113, 227, 453, 906, 909, 1812, 1813, 1818, 2417, 3624, 3626, 3636, 3637, 7248, 7252, 7253, 7272, 7281, 9669, 14496, 14504, 14544, 14549, 14562, 14563, 19338, 28992, 29008, 29013, 29088, 29124, 29125, 30559, 38676, 38677, 38833, 38835, 45839, 54327, 57984

Offset: 1

Author

Alessandro Polcini, Oct 10 2018

Keywords

• nonn

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.

Examples

			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.
		

Crossrefs

Cf. A002450, A006577.

Programs

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;
}

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 *)