A119919 Table read by antidiagonals: number of rationals in [0, 1) having at most n preperiodic bits, then at most k periodic bits (read up antidiagonals).
1, 2, 3, 4, 6, 9, 8, 12, 18, 21, 16, 24, 36, 42, 51, 32, 48, 72, 84, 102, 105, 64, 96, 144, 168, 204, 210, 231, 128, 192, 288, 336, 408, 420, 462, 471, 256, 384, 576, 672, 816, 840, 924, 942, 975, 512, 768, 1152, 1344, 1632, 1680, 1848, 1884, 1950, 1965, 1024
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) = |{ 0/1 = 0.(0)..., 1/3 = 0.(01)..., 2/3 = 0.(10)..., 1/2 = 0.1(0)..., 1/6 = 0.0(01)..., 5/6 = 0.1(10)..., 1/4 = 0.01(0)..., 3/4 = 0.11(0)..., 1/12 = 0.00(01)..., 5/12 = 0.01(10)..., 7/12 = 0.10(01)..., 11/12 = 0.11(10)...}| = 12
Table begins:
1 3 9 21
2 6 18 42
4 12 36 84
8 24 72 168
Programs
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Mathematica
Table[2^n Sum[Plus@@((2^Divisors[j]-1)MoebiusMu[j/Divisors[j]]),{j,1,k}],{n,0,10},{ k,1,10}]
Formula
a(n, k) = 2^n * sum_{j=1..k} sum_{d|j} (2^d - 1) * mu(j/d)