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Numbers with more than one Collatz tripling step whose odd Collatz trajectory does not contain primes.
0

%I #30 May 30 2022 16:33:48

%S 113,227,453,906,909,1812,1813,1818,2417,3624,3626,3636,3637,7248,

%T 7252,7253,7272,7281,9669,14496,14504,14544,14549,14562,14563,19338,

%U 28992,29008,29013,29088,29124,29125,30559,38676,38677,38833,38835,45839,54327,57984

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

%C The starting number itself is not counted in the trajectory, otherwise prime numbers like 113 or 227 wouldn't appear in this sequence.

%C The "odd Collatz trajectory" of a number k is the subset of odd numbers of the full Collatz trajectory of k.

%e 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.

%e 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.

%t 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 *)

%o (Java)

%o for(int i = 0; i < DIM; i++) {

%o if(!collatzAtLeastOnePrime(c) && collatzTriplingSteps(c) > 1)

%o System.out.print(c + ", ");

%o }

%o boolean collatzAtLeastOnePrime(int i) {

%o //first step outside the while loop...

%o if(i % 2 == 0)

%o i /= 2;

%o else

%o i = 3 * i + 1;

%o //...otherwise prime numbers like 113 or 227 would be excluded

%o while(i > 1) {

%o if(i % 2 == 0) {

%o i /= 2;

%o }

%o else {

%o if(BigInteger.valueOf(i).isProbablePrime(10))

%o return true;

%o i = 3 * i + 1;

%o }

%o }

%o return false;

%o }

%Y Cf. A002450, A006577.

%K nonn

%O 1,1

%A _Alessandro Polcini_, Oct 10 2018