A343858 Square array T(m,n), read by ascending antidiagonals. Let f(k) = k/2 if k is even, otherwise ((2*n+1)*k+2*r+1)/2, r is the smallest integer greater than -1, where m = f^j(m) for j > 0 exists and is determined in A345228, T(m,n) is the smallest number reached in the cyclic trajectory of m = f^j(m). f^j(m) means j times recursion into f(m).

0, 0, 1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 1, 1, 1, 0, 1, 1, 3, 1, 5, 0, 1, 1, 3, 1, 5, 3, 0, 1, 1, 3, 1, 5, 3, 7, 0, 1, 1, 3, 1, 5, 3, 7, 1, 0, 1, 1, 3, 1, 1, 3, 7, 1, 9, 0, 1, 1, 3, 1, 5, 3, 7, 1, 3, 5, 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 5, 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 1

Offset

0,10

Comments

This sequence, together with A345228, provides information regarding generalized Collatz functions. (Replace 3*k+1 in the standard Collatz function with a more general a*k+b; then a = 1+2*n and b = 1+2*A345228(m,n).) A345228 tells us which m are part of a cyclic orbit but not if these are part of the same cycle. This sequence identifies each distinct cycle with a different number. Example: If A345228(m1,n) = A345228(m2,n) we know m1 and m2 are part of a cycle but not necessarily the same cycle. If T(m1,n) <> T(m2,n) we know m1 and m2 are not in the same cycle.

The value of n appears to have only a small effect in this sequence and in a majority of cases we find T(m,n) = A000265(m) holds true. This is surprising, given how n is involved in the definition.

Links

Table of n, a(n) for n=0..89.

Thomas Scheuerle, Yellow dot if T(m,n) <> A000265(n); m = 1..100; n = 1..20.

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

Formula

T((1+2*n)*m,n)/T(m,n) = 1+2*n.

T((1+2*(n-b))*m,n)/T(m,n) = 1+2*(n-b). 0 <= b <= n. This formula is only for the majority of cases true if b > 0. For each column m are some rows n where an exception will be seen.

T(m,n) <= A000265(m) (largest odd divisor of m).

T(m,n) = A000265(m) For the majority of all n.

Example

Twelve initial terms of rows 0-10 are listed below:
   n |m->
   0: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 1, 11, ...
   1: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5,  5, ...
   2: 0, 1, 1, 1, 1, 5, 3, 7, 1, 3, 5,  1, ...
   3: 0, 1, 1, 3, 1  5, 3, 7, 1, 9, 5, 11, ...
   4: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, ...
   5: 0, 1, 1, 3, 1, 1, 3, 7, 1, 9, 1, 11, ...
   6: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5,  1, ...
   7: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, ...
   8: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, ...
   9: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, ...
  10: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, ...
Example: T(3,4) = 3 -> f(n): k/2; (9*k+21)/2. This is because r = A345228(3,4) = 10 and 2*10+1 = 21.
f(3) = 24, f(24) = 12, f(12) = 6, f(6) = 3, f(3) = 24, ....
The smallest number in this cycle is 3.

Crossrefs

Cf. A344583, A345228, A000265.

Sequence in context: A119612 A101949 A124796 * A065714 A110700 A338939

Adjacent sequences: A343855 A343856 A343857 * A343859 A343860 A343861

Keyword

nonn,tabl

Author

Thomas Scheuerle, Jun 14 2021

Status

approved