A275866 Number of semiprimes in {n, f(n), f(f(n)), ...., 1}, where f is the Collatz function.

0, 0, 2, 1, 1, 3, 5, 1, 7, 2, 4, 3, 2, 6, 6, 1, 3, 7, 6, 2, 2, 5, 4, 3, 8, 3, 38, 6, 5, 6, 36, 1, 9, 4, 4, 7, 6, 7, 12, 2, 37, 2, 9, 5, 4, 5, 35, 3, 8, 8, 8, 3, 2, 38, 38, 6, 11, 6, 10, 6, 5, 37, 36, 1, 9, 9, 8, 4, 4, 4, 34, 7, 38, 7, 3, 7, 7, 12, 11, 2, 6, 38

Offset: 1

Author

Michel Lagneau, Aug 11 2016

Keywords

• nonn

Comments

Number of semiprimes in the trajectory of n under the 3x+1 map (i.e. the number of semiprimes until the trajectory reaches 1).

It seems that about 15% of the terms satisfy a(i) = a(i+1). For example, up to 100000, 15140 terms satisfy this condition.

Examples

			a(9)=7 because the trajectory of 9 is 9 -> 28 -> 14 -> 7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 and the 7 semiprimes of this trajectory are 9, 14, 22, 34, 26, 10 and 4.
		

Links

• Michel Lagneau, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006577, A078350.

Programs

Mathematica

Table[Count[NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, n, # != 1 &], k_ /; PrimeOmega@ k == 2], {n, 82}] (* Michael De Vlieger, Aug 11 2016 *)

PARI

print1(0, ", ");for(n=2, 100, s=n; t=0; while(s!=1, if(bigomega(s)==2 , t=t+1, t=t); if(s%2==0, s=s/2, s=(3*s+1)); if(s==1, print1(t", "))))