A319227 a(n) is the number of twin primes in the Collatz trajectory of n.

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

Offset: 1

Author

Michel Lagneau, Sep 14 2018

Keywords

• nonn

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 |

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

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

Examples

			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).
		

Links

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

Crossrefs

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

Programs

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: