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

1,7

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 A341021 A358331

Adjacent sequences: A319224 A319225 A319226 * A319228 A319229 A319230

Keyword

nonn

Author

Michel Lagneau, Sep 14 2018

Status

approved