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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
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 |
+----------------+---------------------------------+
...................................................
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Sep 14 2018
STATUS
approved