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A246007 Length of pseudo-Collatz cycles '3*n - 1' of prime numbers. 2
2, 5, 3, 6, 7, 7, 19, 5, 4, 11, 7, 15, 10, 9, 14, 17, 12, 8, 21, 20, 16, 15, 11, 33, 36, 36, 18, 10, 14, 31, 26, 22, 21, 13, 26, 34, 16, 12, 21, 42, 25, 16, 16, 37, 20, 29, 19, 24, 32, 90, 28, 28, 19, 19, 85, 23, 40, 14, 36, 27, 22, 49, 17, 31, 31, 40, 13, 44, 43, 26, 66, 43, 25, 25, 25, 30, 21, 30, 30, 51, 20, 25, 25, 33, 47, 16, 47, 91, 46, 46, 29, 46, 28 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Define a pseudo-Collatz cycle C(prime(n)) = {c(z+1) = c(z)/2 if c(z) mod 2 = 0, otherwise c(z+1) = 3*c(z) - 1}, z >= 1, c(1) = prime(n), n >= 1} depending of the starting point c(1). If c(1) = prime(n) then c(z) might be

  (1) finite convergent to c(z) = 1 or

  (2) infinite periodic from c(z) = 7 or from c(z) = 17 or

  (3) no cycle if c(z) = -1.

The case (3) is not observed out of 10^5 prime numbers. So a(n) = z is the length of the C(prime(n)) up to the stoppping point, where c(z) = 1 or up to the periodical point, where c(z) = 7 or c(z) = 17 or c(z) = c(1). See Table for examples of cases (1) and (2). The longest sequence here is a(99147) = 560 with starting point c(1) = prime(99147) = 1287511 up to the periodical point c(560) = 17.

LINKS

Freimut Marschner, Table of n, a(n) for n = 1..100000

FORMULA

a(n) = z where {c(z+1) = c(z)/2 if c(z) mod 2 = 0, otherwise c(z+1) = 3*c(z) - 1}, z >= 1, c(1) = prime(n), n>= 1}.

EXAMPLE

a(1) = {c(1) = prime(1) = 2, 2 mod 2 = 0, c(2) = 2/2 = 1, z=2} = 2.

Table for cases (1) and (2):

case (1)

c(1) = prime(2) = 3

z    1 2 3 4 5

c(z) 3 8 4 2 1

a(2) = 5

c(1) = prime(3) = 5

z    1  2 3

c(z) 5 14 7

a(3) = 3

c(1) = prime(10) = 29

z     1  2  3   4  5  6  7 8 9 10 11

c(z) 29 86 43 128 64 32 16 8 4  2  1

a(10) = 11

case (2)

c(1) = prime(4) = 7

z    1  2  3 4  5 6  7 ...

c(z) 7 20 10 5 14 7 20 ...

a(4) = 6

c(1) = prime(7) = 17

z     1  2  3  4  5   6  7   8  9 10  11 12  13

c(z) 17 50 25 74 37 110 55 164 82 41 122 61 182

z    14  15  16 17 18 19 20 ...

c(z) 91 272 136 68 34 17 50 ...

a(7) = 19

CROSSREFS

A003627 (Primes of form 3n-1), A006370 (Image of n under the '3x+1' map), A014682 (The Collatz or 3x+1 function: a(n) = n/2 if n is even, otherwise (3n+1)/2), A006577(Number of halving and tripling steps to reach 1 in '3x+1' problem), A016789({3n+2, n >=0} = {3n-1, n >= 1}).

Sequence in context: A264105 A024871 A222072 * A256997 A335499 A239970

Adjacent sequences:  A246004 A246005 A246006 * A246008 A246009 A246010

KEYWORD

nonn

AUTHOR

Freimut Marschner, Aug 10 2014

STATUS

approved

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Last modified September 21 19:57 EDT 2020. Contains 337273 sequences. (Running on oeis4.)