A119918 Table read by antidiagonals: number of rationals in [0, 1) having exactly n preperiodic bits, then exactly k periodic bits (read up antidiagonals).
1, 1, 2, 2, 2, 6, 4, 4, 6, 12, 8, 8, 12, 12, 30, 16, 16, 24, 24, 30, 54, 32, 32, 48, 48, 60, 54, 126, 64, 64, 96, 96, 120, 108, 126, 240, 128, 128, 192, 192, 240, 216, 252, 240, 504, 256, 256, 384, 384, 480, 432, 504, 480, 504, 990, 512, 512, 768, 768, 960, 864, 1008
Offset: 1
Examples
The binary expansion of 7/24 = 0.010(01)... has 3 preperiodic bits (to the right of the binary point) followed by 2 periodic (i.e., repeating) bits, while 1/2 = 0.1(0)... has one bit of each type. The preperiodic and periodic parts are both chosen to be as short as possible.
a(2, 2) = |{1/12 = 0.00(01)..., 5/12 = 0.01(10)..., 7/12 = 0.10(01)..., 11/12 = 0.11(10)...}| = 4
Table begins:
1 2 6 12
1 2 6 12
2 4 12 24
4 8 24 48
Programs
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Mathematica
Table[2^ Max[0,n-1](Plus@@((2^Divisors[k]-1)MoebiusMu[k/Divisors[k]])),{n, 0,1 0},{k,1,10}]
Formula
a(n, k) = 2^max{0, n-1} * sum_{d|k} (2^d - 1) * mu(k/d)