1) Start with a natural number, nNote the conditional statement expressed in steps 2-3. The conjecture, as expressed in step 4, is that such sequences will always terminate in one, independently of the starting value, n_{0}

2) If n is even, then n_{i+1}= n_{i}/2

3) If n is odd, then n_{i+1}= 3n_{i}+ 1

4) Continue at step (2) until n = 1

The 56-cycle Conway sequence, from Richard K. Guy, Tanya Khovanova and Julian Salazar, "Conway's subprime Fibonacci sequences."[3] Shown are the nodes of the sequence, which are coprime odd integers not preceded by an odd term, and the number of terms between them. (Via arXiv Preprint Server)[3]. |

The 136-cycle Conway sequence. Unlike the Collatz sequences, which terminate in one, this sequence is periodic after a few initial terms. (Graph rendered by Gnumeric). |

• A214892 - Conway's subprime Fibonacci sequence starting with (4,1)

• A214893 - Conway's subprime Fibonacci sequence starting with (18, 5)

• A214894 - Conway's subprime Fibonacci sequence starting with (10, 18)

• A214895 - Conway's subprime Fibonacci sequence starting with (23, 162)

• A214896 - Conway's subprime Fibonacci sequence starting with (382, 127)

• A214898 - Conway's subprime Fibonacci sequence, largest loop elements

• A214897 - Conway's subprime Fibonacci sequence : cycle lengths

Need I mention that Conway is best known for The Game of Life, which was a popular pastime when it was introduced in 1970?[4] The principal reason for its popularity is that it could be visualized on the primitive graphical environments of the time.

- Pun intended.
- Jeffrey C. Lagarias, "The 3x + 1 problem and its generalizations," American Mathematical Monthly, vol. 92, no. 1, (January, 1985), pp. 3–23.
- Richard K. Guy, Tanya Khovanova, Julian Salazar, "Conway's subprime Fibonacci sequences," arXiv Preprint Server, July 21, 2012.
- Martin Gardner, "Mathematical Games - The fantastic combinations of John Conway's new solitaire game 'life'," Scientific American, vol. 223 (October 1970), pp. 120-123