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A177000 The Collatz iteration of these primes produces only even numbers, primes and 1. 3
2, 3, 5, 7, 11, 13, 17, 19, 29, 37, 53, 59, 67, 89, 101, 131, 149, 157, 179, 181, 197, 241, 269, 277, 349, 397, 739, 853, 1109, 1237, 1429, 1621, 1861, 1877, 2161, 2389, 2531, 2957, 3413, 3797, 4549, 5717, 7621, 10069, 13397, 17749, 20021, 31541, 40277 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The Collatz iteration of primes of the form (10*4^k-1)/3 produces only one additional prime: 5. The Collatz iteration of primes of the form (13*4^k-1)/3 produces only two additional primes: 5 and 13. This sequence is probably infinite.

In a sense, these are the simplest Collatz iterations starting with a prime number. Except for the increases (3x+1) when an odd prime occurs, the sequence produced by starting with a(n) is decreasing. All the primes that occur in such a Collatz iteration are in this sequence. - T. D. Noe, Oct 05 2011

LINKS

Donovan Johnson and T. D. Noe, Table of n, a(n) for n = 1..10000 (Donovan Johnson to 203 terms)

Eric Weisstein's World of Mathematics, Collatz Problem

Wikipedia, Collatz conjecture

Index entries for sequences related to 3x+1 (or Collatz) problem

EXAMPLE

The Collatz iteration of 7 produces 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, and 1, which are either even, prime, or 1.

MATHEMATICA

Collatz[n_] := NestWhileList[If[EvenQ[ # ], #/2, 3#+1] &, n, #>1 &]; Reap[Do[p=Prime[n]; s=Most[Select[Collatz[p], OddQ]]; If[And@@PrimeQ[s], Sow[p]], {n, PrimePi[10^4]}]][[2, 1]]

oenQ[n_]:=AllTrue[DeleteCases[Most[NestWhileList[If[EvenQ[#], #/2, 3#+1]&, n, #>1&]], _?PrimeQ], EvenQ]; Select[Prime[Range[5000]], oenQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 28 2014 *)

PROG

(Haskell)

a177000 n = a177000_list !! (n-1)

a177000_list = filter (all (\x -> even x || a010051' x == 1) .

                       (init . a070165_row)) a000040_list

-- Reinhard Zumkeller, Apr 03 2012

(PARI) is(n)=isprime(n) && (n<23 || is((3*n+1)>>valuation(3*n+1, 2))) \\ Charles R Greathouse IV, Jun 20 2013

CROSSREFS

Cf. A055509, A078350, A078373, A070165, A010051.

Sequence in context: A061771 A124589 A079150 * A117843 A293667 A068192

Adjacent sequences:  A176997 A176998 A176999 * A177001 A177002 A177003

KEYWORD

nonn,nice

AUTHOR

T. D. Noe, Apr 30 2010

STATUS

approved

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Last modified April 9 14:04 EDT 2020. Contains 333353 sequences. (Running on oeis4.)