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

%I A319227 #7 Sep 28 2018 21:33:28
%S A319227 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,
%T A319227 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,
%U A319227 0,0,0,2,1,2,0,2,1,2,2,0,0,0,0,0,0,2,2
%N A319227 a(n) is the number of twin primes in the Collatz trajectory of n.
%C A319227 Conjecture: a(n) <=2.
%C A319227 For a(n) = 2, the corresponding twin primes are (5, 7) and (11, 13) or (11, 13) and (17, 19).
%C A319227 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).
%C A319227 +----------------+---------------------------------+
%C A319227 | pairs of prime |     pairs of prime numbers      |
%C A319227 |     numbers    |    in the Collatz trajectory    |
%C A319227 |                |    when a(n, p, p+2q) = 2       |
%C A319227 +----------------+---------------------------------+
%C A319227 |    (p, p+2)    |     (5, 7) and (11, 13)         |
%C A319227 |                |  or (11, 13) and (17, 19)       |
%C A319227 +----------------+---------------------------------+
%C A319227 |    (p, p+4)    |     (7, 11) and (13, 17)        |
%C A319227 +----------------+---------------------------------+
%C A319227 |    (p, p+6)    |     (41, 47) and (47, 53)       |
%C A319227 |                |  or (47, 53) and (97, 103)      |
%C A319227 |                |  or (47, 53) and (587, 593)     |
%C A319227 +----------------+---------------------------------+
%C A319227 |    (p, p+8)    |     (23, 31) and (53, 61)       |
%C A319227 +----------------+---------------------------------+
%C A319227 |    (p, p+10)   |     (61, 71) and (73, 83)       |
%C A319227 |                |  or (61, 71) and (283, 293)     |
%C A319227 |                |  or (61, 71) and (577, 587)     |
%C A319227 +----------------+---------------------------------+
%C A319227 |    (p, p+12)   |     (71, 83) and (251, 263)     |
%C A319227 |                |  or (251, 263) and (467, 479)   |
%C A319227 |                |  or (251, 263) and (479, 491)   |
%C A319227 |                |  or (251, 263) and (1607, 1619) |
%C A319227 +----------------+---------------------------------+
%C A319227 |    (p, p+14)   |     No results for n <= 10^6    |
%C A319227 +----------------+---------------------------------+
%C A319227 ...................................................
%H A319227 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%e A319227 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).
%p A319227 nn:=10^8:
%p A319227 for n from 1 to 100 do:
%p A319227    m:=n:lst:={}:
%p A319227       for i from 1 to nn while(m<>1) do:
%p A319227         if irem(m, 2)=0
%p A319227          then
%p A319227          m:=m/2:
%p A319227          else
%p A319227          lst:=lst union {m}:m:=3*m+1:
%p A319227        fi:
%p A319227      od:
%p A319227      n0:=nops(lst):it:=0:
%p A319227      for j from 1 to n0-1 do:
%p A319227      if isprime(lst[j]) and isprime(lst[j+1]) and lst[j+1]=lst[j]+2
%p A319227      then it:=it+1:else fi:
%p A319227       od:
%p A319227     printf(`%d, `,it):
%p A319227     od:
%Y A319227 Cf. A006370, A070165, A078350, A196871, A280408.
%K A319227 nonn
%O A319227 1,7
%A A319227 _Michel Lagneau_, Sep 14 2018