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A319936 Numbers n with more than one Collatz tripling steps whose odd Collatz trajectory does not contain primes. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The starting number n 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 n is the subset of odd numbers of the full Collatz trajectory of n.

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

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Last modified September 20 16:50 EDT 2021. Contains 347586 sequences. (Running on oeis4.)