A181921 -id:A181921 - cp's OEIS frontend

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

Joseph L. Pe, Dec 23 2002

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

			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.

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

Cf. A064684, A006370, A010051.

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

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

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 *)

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

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

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

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

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

A078373 n sets a record for the 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.

Original entry on oeis.org

2, 3, 7, 19, 27, 97, 171, 231, 487, 763, 1071, 4011, 6171, 10971, 17647, 47059, 99151, 117511, 202471, 260847, 481959, 963919, 1564063, 1805311, 1993215, 6991599, 8400511, 11200681, 36791535, 46564287, 103359483, 206718967, 359502063

Offset: 1

Joseph L. Pe, Dec 24 2002

nonn

			The sequence n, f(n), f(f(n)), ..., 1 for n = 7 is: 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, which has six prime terms, more prime terms than for any n < 7. Hence 7 sets a record and so belongs to the sequence.

Jud McCranie, Table of n, a(n) for n = 1..40
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
Carlos Rivera, Puzzle 634. Primes in Collatz trajectory
Carlos Rivera, Puzzle 922. Follow up to Puzzle 634

A362958 gives the corresponding numbers of primes.

Cf. A055509, A181921.

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]; high = 0; a = {}; For[j = 1, j <= 10^5, j++, k = g[j]; If[k > high, high = k; a = Append[a, j]]]; a
(* Second program: *)
With[{s = Array[Count[NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, #, # > 1 &], ?PrimeQ] &, 10^5]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* _Michael De Vlieger, Apr 21 2018 *)

A078350(n)=my(s=isprime(n)); while(n>1, if(n%2, n=(3*n+1)/2, n/=2); s+=isprime(n)); s
r=0; for(n=2,1e9, t=A078350(n); if(t>r, r=t; print1(n", "))) \\ Charles R Greathouse IV, Apr 28 2015

a(18)-a(30) from Donovan Johnson, Jul 02 2010

a(31)-a(33) from Carlos Rivera, Apr 15 2012

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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.

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A078373 n sets a record for the 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.

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Programs

Mathematica

PARI

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