A284904 The number of integers k less than 2^n whose decimal representation of their reciprocals has an odd period.

0, 0, 1, 2, 5, 10, 20, 40, 80, 150, 280, 522, 965, 1802, 3393, 6426, 12197, 23236, 44419, 85116, 163543, 314837, 607440, 1174134, 2273619, 4409116, 8561931, 16646790, 32404446, 63145533, 123173667

Offset: 0

Author

Robert G. Wilson v, Apr 05 2017

Keywords

• nonn

Comments

The number of terms in A284601 less than 2^n.

Examples

			a(5)=10 because the reciprocals of {3, 6, 9, 12, 15, 18, 24, 27, 30, 31} all have an odd period, i.e., {1, 1, 1, 1, 1, 1, 1, 3, 1, 15}, respectively.
		

Links

• Index entries for sequences related to decimal expansion of 1/n

Crossrefs

Cf. A284601.

Programs

Mathematica

f[n_] := Mod[ Length[ RealDigits[1/n][[1, -1]]], 2]; s = 0; k = 1; lst = {}; Do[ While[k < 2^n, s += f@k; k++]; AppendTo[lst, s], {n, 0, 18}] (* or *)
g[n_] := Mod[ MultiplicativeOrder[10, FixedPoint[ Quotient[#, GCD[#, 10]] &, n]], 2]; h[n_] := Length@ Most@ Flatten@ Table[2^i*5^j, {i, 0, n}, {j, 0, Log[5, 2^(n -i)]}]; s = 0; k = 1; lst = {}; Do[ While[k < 2^n, s += g@k; k++]; AppendTo[lst, s - h[n]], {n, 0, 30}]

Formula

log(a(n)) is approximately 0.33 + 0.727*n - 0.643*sqrt(n).