A267830 Number of nonprime numbers in {n, f(n), f(f(n)), ...., 1}, where f is the Collatz function defined by f(x) = x/2 if x is even; f(x) = 3x + 1 if x is odd.

1, 1, 5, 2, 4, 6, 11, 3, 14, 5, 10, 7, 7, 12, 14, 4, 9, 15, 14, 6, 7, 11, 12, 8, 17, 8, 87, 13, 13, 15, 83, 5, 20, 10, 11, 16, 15, 15, 24, 7, 85, 8, 22, 12, 13, 13, 82, 9, 18, 18, 19, 9, 9, 88, 89, 14, 25, 14, 22, 16, 15, 84, 88, 6, 21, 21, 19, 11, 12, 12, 81

Offset: 1

Author

Michel Lagneau, Jan 21 2016

Keywords

• nonn

Comments

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

It seems that about 20% of the terms satisfy a(i) = a(i+1). For example, up to 10^6, 201085 terms satisfy this condition.

Examples

			a(9)=14 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 14 nonprimes of this trajectory are 9, 28, 14, 22, 34, 52, 26, 40, 20, 10, 16, 8, 4, and 1.
		

Links

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

Crossrefs

Cf. A006577, A008908, A078350, A018252.

Programs

Mathematica

A267830[n_] := Count[NestWhileList[If[EvenQ@#, #/2, 3 # + 1] &, n, # != 1 &], ?(Not@PrimeQ@# &)] (* _JungHwan Min, Jan 24 2016 *)

PARI

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

Formula

a(n)= A008908(n) - A078350(n).