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

1,3

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.

Links

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

Example

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.

Mathematica

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

Prog

(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", "))))

Crossrefs

Cf. A006577, A078350.

Sequence in context: A230698 A090234 A286380 * A007754 A144866 A058732

Adjacent sequences: A275863 A275864 A275865 * A275867 A275868 A275869

Keyword

nonn

Author

Michel Lagneau, Aug 11 2016

Status

approved