A395539 - OEIS

%I #19 May 06 2026 14:14:13

%S 0,89,178,267,356,445,534,1,712,801,890,1,1068,59,2,1335,1424,475,

%T 1602,83,1780,3,2,2047,2136,2225,118,2403,4,71,2670,67,2848,3,950,5,

%U 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

%N Successor function of Conway's PIGAME.

%C 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.)

%C 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.

%H Pontus von Brömssen, <a href="/A395539/b395539.txt">Table of n, a(n) for n = 0..10000</a>

%H John H. Conway, <a href="https://doi.org/10.1007/978-1-4612-4808-8_2">FRACTRAN: A Simple Universal Programming Language for Arithmetic</a>. 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.)

%H Richard K. Guy, <a href="https://doi.org/10.2307/2690263">Conway's Prime Producing Machine</a>, Mathematics Magazine, Vol. 56, No. 1 (1983), pp. 26-33, available at JSTOR.org/stable/2690263.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/FRACTRAN">FRACTRAN</a>, created Sep. 23, 2007.

%e 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.

%e Similarly, for n = 2, 3, 4, 5 and 6, there's no denominator > 1 that divides n, therefore we get a(n) = n*89.

%e 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).

%e For n = 13, we find denominator 13 = A350556(34) that divides n, whence a(13) = 13/13*A350555(34) = 59.

%e For n = 14, we find again denominator 7 = A350556(36) which divides n, and with A350555(36) = 1, a(14) = 14/7*1 = 2.

%e 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]

%o (Python)

%o 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

%o 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

%o A395539=lambda x: next(x//d*n for n, d in zip(A350555, A350556) if x%d==0)

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

%Y Cf. A203907 (similar for PRIMEGAME).

%Y Cf. A000796.

%K nonn,easy

%O 0,2

%A _M. F. Hasler_, Apr 27 2026