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