A284601 -id:A284601 - cp's OEIS frontend

A284602 Numbers k such that the decimal representation of 1/k is either finite or has even period.

1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 32, 33, 34, 35, 38, 39, 40, 42, 44, 46, 47, 49, 50, 51, 52, 55, 56, 57, 58, 59, 61, 63, 64, 65, 66, 68, 69, 70, 73, 76, 77, 78, 80, 84, 85, 87, 88, 89, 91, 92, 94, 95, 97, 98, 99, 100, 101, 102, 103, 104, 105, 109, 110, 112, 113, 114, 115

Offset: 1

Ilya Gutkovskiy, Mar 30 2017

All numbers of the form 2^i*5^j with i, j >= 0 are in this sequence (numbers with a finite decimal expansion).
From Robert G. Wilson v, Apr 02 2017: (Start)
If k is in the sequence, then so are 2k and 5k.
The complement of A284601.
Primitives: 1, 7, 11, 13, 17, 19, 21, 23, 29, 33, 39, 47, 49, 51, 57, 59, 61, 63, ..., .
(End)

			14 is in the sequence because 1/14 = 0.0714285(714285)..., whose period is 6, an even number.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Index entries for sequences related to decimal expansion of 1/n

Cf. A002371, A003592, A028416, A051626, A206586, A284601.

Select[Range[115], Mod[Length[RealDigits[1/#][[1, -1]]], 2] == 0 & ]

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

Original entry on oeis.org

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

Robert G. Wilson v, Apr 05 2017

nonn

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

			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.

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

Cf. A284601.

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}]

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

cp's OEIS Frontend

A284602 Numbers k such that the decimal representation of 1/k is either finite or has even period.

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