A320227 Assuming the truth of the Collatz conjecture, let T(n,i), i = 1..k be the initial k elements of the Collatz trajectory of n, up to when the first 1 appears, but excluding the 1. a(n) is the number of ordered pairs T(n,i) < T(n,j) such that gcd(T(n,i), T(n,j)) = 1.

0, 0, 10, 0, 4, 11, 58, 0, 84, 4, 40, 12, 12, 62, 47, 0, 25, 89, 89, 4, 6, 43, 36, 13, 117, 13, 3395, 66, 66, 49, 3064, 0, 148, 27, 21, 94, 94, 94, 286, 4, 3246, 6, 184, 46, 42, 39, 2924, 14, 122, 122, 120, 14, 14, 3435, 3374, 70, 231, 70, 247, 51, 63, 3101

Offset: 1

Author

Michel Lagneau, Oct 08 2018

Keywords

• nonn

Comments

If the number 1 of the Collatz trajectory is included, we obtain the new sequence b(n) = a(n) + A006577(n).

We observe interesting properties for the even and odd values of a(n).

First case: a(n) = 0, 4, 6, ..., 2i, ...

When a(n) = q even, there exists a subset N(q) = {n_1, n_2, ...} such that a(n_i) = q for i = 1, 2, ... We observe that N(q) = N1(q) union N2(q) (see the table below). Conjecturally, for n = 12, 14, 16, ... N1(q) is finite and the last two elements of the set N1(q) are of the form x and x+1.

The elements of N2(q) are of the form {((4^m - 1)/3)*2^k}, k = 0, 1, ... with m = a(n)/2. The set N2(q) is infinite.

Second case: a(n) = 11, 13, 15, ...

Conjecturally, N1(q) is finite and the last two elements of the set N1(q) are of the form y and y+2.

Conjecture: N2(q) = { }.

The following table gives the first 17 values of a(n) in ascending order with the corresponding subsets N1(q) and N2(q).

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

|a(n)| N1(a(n)) |

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

| 0 |{ } |

| 4 |{ } |

| 6 |{ } |

| 8 |{ } |

| 10 |{3} |

| 11 |{6} |

| 12 |{12, 13} |

| 13 |{24, 26} |

| 14 |{48, 52, 53} |

| 15 |{96, 104, 106} |

| 16 |{192, 208, 212, 213} |

| 17 |{384, 416, 424, 426} |

| 18 |{768, 832, 848, 852, 853} |

| 19 |{113, 1536, 1664, 1696, 1704, 706} |

| 20 |{226, 3072, 3328, 3392, 3408, 3412, 3413} |

| 21 |{35, 452, 453, 6144, 6656, 6784, 6816, 6824, 6826} |

| 22 |{70, 227, 904, 906, 12288, 13312, 13568, 13632, 13648, 13652, 13653}|

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

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

|a(n)| N2(a(n)) |

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

| 0 |{1, 2, 4, 8, 16, 32, ..., 2^k, ... } (A000079) |

| 4 |{5, 10, 20, 40, 80, ..., 5*2^k, ...} (A020714) |

| 6 |{21, 42, 84, 168, 336, 672, ..., ((4^3 - 1)/3)*2^k, ...} (A175805) |

| 8 |{85, 170, 340, 680, ..., ((4^4 - 1)/3)*2^k, ...} |

| 10 |{341, 682, 1364, 2728, ..., ((4^5 - 1)/3)*2^k, ...} |

| 11 | { } |

| 12 |{1365, 2730, 5460, ...,((4^6 - 1)/3)*2^k, ...} |

| 13 | { } |

| 14 |{5461, 10922, ..., ((4^7 - 1)/3)*2^k, ...} |

| 15 | { } |

| 16 |{21845, 43690, ...,((4^8 - 1)/3)*2^k, ...} |

| 17 | { } |

| 18 |{87381, 174762, ...,((4^9 - 1)/3)*2^k, ...} |

| 19 | { } |

| 20 |{349525, 699050, ..., ((4^10 - 1)/3)*2^k, ...} |

| 21 | { } |

| 22 |{1398101, 2796202, ..., ((4^11 - 1)/3)*2^k, ...} |

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

Examples

			a(3) = 10 because the Collatz trajectory T(3,i) of 3 up to the number 1 is 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2  and gcd(T(i), T(j)) = 1 for the 10 following pairs of elements of T: (2, 3), (2, 5), (3, 4), (3, 5), (3, 8), (3, 10), (3, 16), (4, 5), (5, 8) and (5, 16). 28
In the general case, a(n) = 10 for n in the set {3} union {341, 682, 1364, 2728, ...,((4^5 - 1)/3)*2^k, ...} with k = 0, 1, 2, ...
a(6) = 11 because the Collatz trajectory T(6,i) of 6 up to the number 1 is 6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2  and gcd(T(i), T(j)) = 1 for the 11 following pairs of elements of T: (2, 3), (2, 5), (3, 4), (3, 5), (3, 8), (3, 10), (3, 16), (4, 5), (5, 6), (5, 8) and (5, 16).
		

Links

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

Crossrefs

Cf. A000079, A002450, A006370, A006577, A020714, A175805.

Programs

Maple

nn:=1000:
for n from 1 to 200 do:
   m:=n:lst:={}:
      for i from 1 to nn while(m<>1) do:
        if irem(m, 2)=0
         then
         lst:=lst union {m}: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:
      for k from j+1 to n0 do:
       if gcd(lst[j],lst[k])=1
       then
        it:=it+1:
        else fi:
    od:
    od:
  printf(`%d, `,it):
od:

Extensions

Definition revised by N. J. A. Sloane, Nov 12 2018