A395539 Successor function of Conway's PIGAME.

0, 89, 178, 267, 356, 445, 534, 1, 712, 801, 890, 1, 1068, 59, 2, 1335, 1424, 475, 1602, 83, 1780, 3, 2, 2047, 2136, 2225, 118, 2403, 4, 71, 2670, 67, 2848, 3, 950, 5, 3204, 23, 13, 177, 3560, 86, 6, 53, 4, 4005, 365, 833, 4272, 7, 4450, 1425, 236, 41, 4806, 5, 8, 89, 142, 19, 5340, 37, 134

Offset

0,2

Comments

a(n) is the first integer in the sequence (n*A350555(k)/A350556(k); k=1..40). (Equivalently, k is the first index for which the denominator A350556(k) divides n.)

It yields the k-th digit of the decimal expansion of Pi, A000796(1+k), as binary logarithm of the first power of two in the trajectory of 89*2^k under iterations of this map. That is, the first power of two that appears in each of these trajectories equals 2^r(Pi*10^k), with r(x) = floor(x) mod 10 = last digit of the integer part of x.

Links

Pontus von Brömssen, Table of n, a(n) for n = 0..10000

John H. Conway, FRACTRAN: A Simple Universal Programming Language for Arithmetic. In: Cover, T.M., Gopinath, B. (Eds), Open Problems in Communication and Computation. Springer, New York, 1987. doi:10.1007/978-1-4612-4808-8_2. (Incorrectly states that the starting value to get pi(n) is 2^n.)

Richard K. Guy, Conway's Prime Producing Machine, Mathematics Magazine, Vol. 56, No. 1 (1983), pp. 26-33, available at JSTOR.org/stable/2690263.

Wikipedia, FRACTRAN, created Sep. 23, 2007.

Example

For n = 1, there is no denominator > 1 in A350556 that may divide n, so we get to the last index 40 where A350556(40) = 1, and multiply n with the 40th numerator, A350555(40) = 89, so we get a(1) = 89.
Similarly, for n = 2, 3, 4, 5 and 6, there's no denominator > 1 that divides n, therefore we get a(n) = n*89.
For n = 7, we find denominator 7 = A350556(36) which divides n, and since A350555(36) = 1, we have a(7) = 7/7*1 = 1; similarly for n = 11 = A350556(37).
For n = 13, we find denominator 13 = A350556(34) that divides n, whence a(13) = 13/13*A350555(34) = 59.
For n = 14, we find again denominator 7 = A350556(36) which divides n, and with A350555(36) = 1, a(14) = 14/7*1 = 2.
The trajectory of 89 under this map is 89, 517, 9163, 256025, 20825, 581875, 2541875, 3399375, 2113125, 380625, 931875, 1674750, 4100250, ..., where 8 = 2^3 is the first power of two to occur, after 775 iterations, corresponding to the initial digit of Pi. [Corrected by Pontus von Brömssen, May 04 2026]

Prog

(Python)
A350555=365, 29, 79, 679, 3159, 83, 473, 638, 434, 89, 17, 79, 31, 41, 517, 111, 305, 23, 73, 61, 37, 19, 89, 41, 833, 53, 86, 13, 23, 67, 71, 83, 475, 59, 41, 1, 1, 1, 1, 89 # PIGAME numerators
A350556=46, 161, 575, 451, 413, 407, 371, 355, 335, 235, 209, 122, 183, 115, 89, 83, 79, 73, 71, 67, 61, 59, 57, 53, 47, 43, 41, 38, 37, 31, 29, 19, 17, 13, 291, 7, 11, 1024, 97, 1 # PIGAME denominators
A395539=lambda x: next(x//d*n for n, d in zip(A350555, A350556) if x%d==0)

Crossrefs

Cf. A350555, A350556 (numerators and denominators of PIGAME).

Cf. A203907 (similar for PRIMEGAME).

Cf. A000796.

Sequence in context: A044259 A044640 A260868 * A106758 A142335 A230168

Adjacent sequences: A395536 A395537 A395538 * A395540 A395542 A395543

Keyword

nonn,easy

Author

M. F. Hasler, Apr 27 2026

Status

approved