A181921 The smallest positive integer that produces exactly n primes in a Collatz trajectory.

2, 5, 3, 15, 11, 7, 19, 43, 67, 89, 39, 127, 123, 223, 111, 351, 175, 155, 103, 63, 107, 71, 47, 31, 27, 97, 193, 171, 231, 487, 1087, 763, 2223, 2143, 1263, 1071, 4011, 6919, 8127, 13183, 6591, 6943, 6171, 10971, 46443, 48927, 35295, 17647, 70589, 47059

Offset: 1

Author

G. L. Honaker, Jr., Apr 01 2012

Keywords

• nonn

Examples

			a(6) = 7 because the Collatz trajectory of 7 is {7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1}, containing 6 primes {7, 11, 17, 13, 5, 2}, and 7 is the smallest positive integer for which exactly 6 primes occur via this trajectory.
		

Links

• Reinhard Zumkeller and Jud McCranie, Table of n, a(n) for n = 1..92 (first 75 numbers from Reinhard Zumkeller)
• Eric Weisstein's World of Mathematics, Collatz Problem
• Wikipedia, Collatz conjecture
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A055509, A078350.

Programs

Haskell

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a181921 = (+ 1) . fromJust . (`elemIndex` a078350_list)
-- Reinhard Zumkeller, Apr 03 2012

Mathematica

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; nn = 50; t = Table[0, {nn}]; t[[1]] = 2; todo = nn - 1; n = 3; While[todo > 0, ps = Length[Select[Collatz[n], PrimeQ]]; If[ps <= nn && t[[ps]] == 0, t[[ps]] = n; todo--]; n = n + 2]; t (* T. D. Noe, Apr 02 2012 *)
With[{lst=Table[Count[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&],?PrimeQ],{n, 71000}]},Table[Position[lst,k,1,1],{k,50}]//Flatten] (* _Harvey P. Dale, Sep 08 2018 *)

PARI

np(n)=my(t=1);while(n>2,t+=isprime(n);if(n%2,n+=n>>1+1,n>>=1));t
v=vector(40);n=1;while(1,t=np(n++);if(t<=#v&&v[t]==0, v[t]=n; if(vecmin(v), return(v)))) \\ Charles R Greathouse IV, Apr 01 2012

Extensions

a(13)-a(50) from Charles R Greathouse IV, Apr 01 2012