OFFSET
1,5
COMMENTS
For any n > 0 and k > 0, if the Collatz conjecture is true, then:
- A006370^i(n) = 1 and A006370^j(k) = 1 for some i >= 0 and j >= 0 (where A006370^i denotes the i-th iterate of A006370; actually i = A006577(n) and j = A006577(k)),
- hence the Collatz sequences starting at n and k meet,
- let c be the greatest number between 0 and min(i, j) inclusive such that A006370^(i-c)(n) = A006370^(j-c)(k),
- then T(n, k) = A006370^(i-c)(n).
LINKS
FORMULA
For any m > 0, n > 0 and k > 0, and provided that the Collatz conjecture is true:
- T(n, n) = n,
- T(n, k) = T(k, n) (T is commutative),
- T(m, T(n, k)) = T(T(m, n), k) (T is associative),
- T(n, 1) = 1 (1 is an absorbing element for T),
- T(n * 2^k, n) = n,
- T(n, k) = 1 iff n = 1 or k = 1.
EXAMPLE
For T(12, 13):
- The Collatz sequence starting at 12 is: 12, 6, 3, 10, 5, 16, 8, 4, 2, 1,
- The Collatz sequence starting at 13 is: 13, 40, 20, 10, 5, 16, 8, 4, 2, 1,
- They meet at the value 10, hence T(12, 13) = 10.
Array T(n, k) begins:
n\k| 1 2 3 4 5 6 7 8 9 10
---+--------------------------------------------------
1| 1 1 1 1 1 1 1 1 1 1
2| 1 2 2 2 2 2 2 2 2 2
3| 1 2 3 4 5 3 10 8 10 10
4| 1 2 4 4 4 4 4 4 4 4
5| 1 2 5 4 5 5 5 8 5 5
6| 1 2 3 4 5 6 10 8 10 10
7| 1 2 10 4 5 10 7 8 7 10
8| 1 2 8 4 8 8 8 8 8 8
9| 1 2 10 4 5 10 7 8 9 10
10| 1 2 10 4 5 10 10 8 10 10
PROG
(PARI) T(n, k) = my (nn=[]); while (1, nn = concat(nn, n); if (n==1, break); n=if (n%2, 3*n+1, n/2)); nn=Set(nn); while (!setsearch(nn, k), k=if (k%2, 3*k+1, k/2)); k
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Rémy Sigrist, Feb 25 2018
STATUS
approved