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A319936
Numbers with more than one Collatz tripling step 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
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
Sequence in context: A142426 A309617 A210512 * A142700 A142002 A142917
KEYWORD
nonn
AUTHOR
Alessandro Polcini, Oct 10 2018
STATUS
approved