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

1,1

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.

Links

Table of n, a(n) for n=1..40.

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

Cf. A002450, A006577.

Sequence in context: A142426 A309617 A210512 * A142700 A142002 A142917

Adjacent sequences: A319933 A319934 A319935 * A319937 A319938 A319939

Keyword

nonn

Author

Alessandro Polcini, Oct 10 2018

Status

approved