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A300119 Square array T(n, k) read by antidiagonals, n > 0 and k > 0: T(n, k) is the value where Collatz sequences starting at n and at k meet. 1
1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 2, 4, 4, 2, 1, 1, 2, 5, 4, 5, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 10, 4, 5, 4, 10, 2, 1, 1, 2, 8, 4, 5, 5, 4, 8, 2, 1, 1, 2, 10, 4, 5, 6, 5, 4, 10, 2, 1, 1, 2, 10, 4, 8, 10, 10, 8, 4, 10, 2, 1, 1, 2, 10, 4, 5, 8 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..84.

Rémy Sigrist, Scatterplot of (n, k) such that T(n, k) = 40 and 1 <= n <= 1000 and 1 <= k <= 1000

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

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

Cf. A006370, A006577.

Sequence in context: A003983 A087062 A204026 * A323211 A110537 A144434

Adjacent sequences:  A300116 A300117 A300118 * A300120 A300121 A300122

KEYWORD

nonn,tabl

AUTHOR

Rémy Sigrist, Feb 25 2018

STATUS

approved

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Last modified February 20 22:47 EST 2020. Contains 332086 sequences. (Running on oeis4.)