A386406 Length of the preperiodic part of the decimal expansion of 1/n, including any leading zeros from the period.

1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 3

Offset

2,3

Comments

See A114205.

Links

Table of n, a(n) for n=2..92.

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

Formula

a(n) = p + (floor(log_10(1/f)) if f!=0), where p = A051628(n) and f = frac(10^p/n). - Kevin Ryde, Jul 22 2025

Example

For n = 92, 1/n = 0.01(0869565217391304347826) = 0.010(8695652173913043478260), so the preperiodic part is "010" and has length a(92) = 3.

Mathematica

b[n_] := Block[{p, o}, {p, o} = RealDigits[1/n]; If[!IntegerQ[Last[p]], p = Join[Most[p], TakeWhile[Last[p], #==0&]]]; Length[p]-o];
Table[b[n], {n, 2, 100}]

Prog

(PARI) a(n) = my(pre = max(valuation(n, 2), valuation(n, 5)), r = 10^pre % n); pre + if(r, logint(n\r, 10)); \\ Kevin Ryde, Jul 22 2025

Crossrefs

Cf. A051628, A114205, A114206.

Sequence in context: A127967 A147602 A114206 * A073202 A337345 A294904

Adjacent sequences: A386403 A386404 A386405 * A386407 A386408 A386409

Keyword

nonn,base,easy

Author

Andrei Zabolotskii, Jul 20 2025

Status

approved