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

1,3

Links

Table of n, a(n) for n=1..62.

Formula

a(n, k) = 2^max{0, n-1} * sum_{d|k} (2^d - 1) * mu(k/d)

Example

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

Mathematica

Table[2^ Max[0, n-1](Plus@@((2^Divisors[k]-1)MoebiusMu[k/Divisors[k]])), {n, 0, 1 0}, {k, 1, 10}]

Crossrefs

Outer product of A011782 and A038199.

Sequence in context: A210550 A208659 A209752 * A084867 A194949 A227550

Adjacent sequences: A119915 A119916 A119917 * A119919 A119920 A119921

Keyword

nonn,base,easy,tabl

Author

Brad Chalfan (brad(AT)chalfan.net), May 28, 2006

Status

approved