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

0,4

Comments

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

Links

Table of n, a(n) for n=0..30.

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

Formula

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

Example

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.

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}]

Crossrefs

Cf. A284601.

Sequence in context: A327287 A296122 A293324 * A084215 A024810 A049938

Adjacent sequences: A284901 A284902 A284903 * A284905 A284906 A284907

Keyword

nonn

Author

Robert G. Wilson v, Apr 05 2017

Status

approved