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A319227 a(n) is the number of twin primes in the Collatz trajectory of n. 0
0, 0, 1, 0, 0, 1, 2, 0, 2, 0, 1, 1, 0, 2, 0, 0, 0, 2, 2, 0, 0, 1, 0, 1, 2, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 0, 2, 1, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 2, 2, 1, 2, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 2, 1, 2, 0, 2, 1, 2, 2, 0, 0, 0, 0, 0, 0, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

Conjecture: a(n) <=2.

For a(n) = 2, the corresponding twin primes are (5, 7) and (11, 13) or (11, 13) and (17, 19).

This sequence is generalizable: let a(n, p, p+2q) be the number of pairs of primes of form (p, p+2q) in the Collatz trajectory of n, q = 1, 2,... It is conjectured that a(n, p, p+2q) < =2. (see the table below).

+----------------+---------------------------------+

| pairs of prime |     pairs of prime numbers      |

|     numbers    |    in the Collatz trajectory    |

|                |    when a(n, p, p+2q) = 2       |

+----------------+---------------------------------+

|    (p, p+2)    |     (5, 7) and (11, 13)         |

|                |  or (11, 13) and (17, 19)       |

+----------------+---------------------------------+

|    (p, p+4)    |     (7, 11) and (13, 17)        |

+----------------+---------------------------------+

|    (p, p+6)    |     (41, 47) and (47, 53)       |

|                |  or (47, 53) and (97, 103)      |

|                |  or (47, 53) and (587, 593)     |

+----------------+---------------------------------+

|    (p, p+8)    |     (23, 31) and (53, 61)       |

+----------------+---------------------------------+

|    (p, p+10)   |     (61, 71) and (73, 83)       |

|                |  or (61, 71) and (283, 293)     |

|                |  or (61, 71) and (577, 587)     |

+----------------+---------------------------------+

|    (p, p+12)   |     (71, 83) and (251, 263)     |

|                |  or (251, 263) and (467, 479)   |

|                |  or (251, 263) and (479, 491)   |

|                |  or (251, 263) and (1607, 1619) |

+----------------+---------------------------------+

|    (p, p+14)   |     No results for n <= 10^6    |

+----------------+---------------------------------+

...................................................

LINKS

Table of n, a(n) for n=1..87.

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

EXAMPLE

a(7) = 2 because the Collatz trajectory of 7 is 7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 with two twin primes: (5, 7) and (11, 13).

MAPLE

nn:=10^8:

for n from 1 to 100 do:

   m:=n:lst:={}:

      for i from 1 to nn while(m<>1) do:

        if irem(m, 2)=0

         then

         m:=m/2:

         else

         lst:=lst union {m}:m:=3*m+1:

       fi:

     od:

     n0:=nops(lst):it:=0:

     for j from 1 to n0-1 do:

     if isprime(lst[j]) and isprime(lst[j+1]) and lst[j+1]=lst[j]+2

     then it:=it+1:else fi:

      od:

    printf(`%d, `, it):

    od:

CROSSREFS

Cf. A006370, A070165, A078350, A196871, A280408.

Sequence in context: A226786 A266330 A320313 * A219490 A277142 A240592

Adjacent sequences:  A319224 A319225 A319226 * A319228 A319229 A319230

KEYWORD

nonn

AUTHOR

Michel Lagneau, Sep 14 2018

STATUS

approved

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Last modified July 5 20:21 EDT 2020. Contains 335473 sequences. (Running on oeis4.)