A078350 Number of primes 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.

0, 1, 3, 1, 2, 3, 6, 1, 6, 2, 5, 3, 3, 6, 4, 1, 4, 6, 7, 2, 1, 5, 4, 3, 7, 3, 25, 6, 6, 4, 24, 1, 7, 4, 3, 6, 7, 7, 11, 2, 25, 1, 8, 5, 4, 4, 23, 3, 7, 7, 6, 3, 3, 25, 24, 6, 8, 6, 11, 4, 5, 24, 20, 1, 7, 7, 9, 4, 3, 3, 22, 6, 25, 7, 2, 7, 6, 11, 11, 2, 5, 25, 24, 1, 1, 8, 9, 5, 10, 4, 20, 4, 3, 23, 20

Offset: 1

Author

Joseph L. Pe, Dec 23 2002

Keywords

• nonn
• nice

Comments

Number of primes in the trajectory of n under the 3x+1 map (i.e., the number of primes until the trajectory reaches 1, including 2 once). - Benoit Cloitre, Dec 23 2002

a(A196871(n)) = 0. - Reinhard Zumkeller, Oct 08 2011

a(A181921(n)) = n and a(m) <> n for m < A181921(n). - Reinhard Zumkeller, Apr 03 2012

Examples

			3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1; in this trajectory 3, 5, 2 are primes hence a(3) = 3. - _Benoit Cloitre_, Dec 23 2002
The finite sequence n, f(n), f(f(n)), ..., 1 for n = 12 is 12, 6, 3, 10, 5, 16, 8, 4, 2, 1, which has three prime terms. Hence a(12) = 3.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
• Eric Weisstein's World of Mathematics, Collatz Problem
• Wikipedia, Collatz conjecture
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A064684, A006370, A010051.

Programs

Haskell

a078350 n = sum $ map a010051 $ takeWhile (> 1) $ iterate a006370 n  -- Reinhard Zumkeller, Oct 08 2011

Maple

a:= proc(n) option remember; `if`(n=1, 0,
     `if`(isprime(n), 1, 0)+a(`if`(n::even, n/2, 3*n+1)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 04 2024

Mathematica

f[n_] := n/2 /; Mod[n, 2] == 0 f[n_] := 3 n + 1 /; Mod[n, 2] == 1 g[n_] := Module[{i, p}, i = n; p = 0; While[i > 1, If[PrimeQ[i], p = p + 1]; i = f[i]]; p]; Table[g[n], {n, 1, 100}]
Table[Count[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#!=1&],?PrimeQ],{n,100}] (* _Harvey P. Dale, Aug 29 2012 *)

PARI

for(n=2,500,s=n; t=0; while(s!=1,if(isprime(s)==1,t=t+1,t=t); if(s%2==0,s=s/2,s=(3*s+1)); if(s==1,print1(t,","); ); )) \\ Benoit Cloitre, Dec 23 2002

PARI

a(n)=my(s=isprime(n));while(n>1,if(n%2,n=(3*n+1)/2,n/=2);s+=isprime(n));s \\ Charles R Greathouse IV, Apr 28 2015

PARI

A078350(n,c=n>1)={while(1>=valuation(n,2), isprime(n)&&c++; n=n*3+1);c} \\ M. F. Hasler, Dec 05 2017

Formula

a(n) = A055509(n) + 1 for n > 1.

a(n) = 1 when n > 1 is in A000079, i.e., a power of 2. - Benoit Cloitre, Dec 20 2017

Extensions

Edited by N. J. A. Sloane, Jan 17 2009 at the suggestion of R. J. Mathar