A100491 Periodicity of the reciprocal of the Mersenne numbers (A001348).

1, 6, 15, 42, 44, 1365, 3855, 74898, 44620, 39672, 195225786, 616318176, 26815350376, 1186422030, 663226400, 1001874900, 4885233465012400, 1152921504606846975, 10197205073773110, 9758202933231640, 2908370863958880, 1152589154156603558, 2895630705663782454386, 103161669940448356241593685

Offset: 1

Author

Robert G. Wilson v, Nov 22 2004

Keywords

• base
• nonn

Comments

"The answer to the second question, as to whether there is a point beyond which all primes yield periods shorter than p-1, is unknown. It is widely believed that there should be infinitely many primes for which the period is exactly p-1, but at present we cannot be certain." [Ball]

References

• Keith Ball, Strange Curves, Counting Rabbits and other Mathematical Explorations, Princeton University Press, Princeton and Oxford, 2003, Page 57.

Crossrefs

Cf. A001348.

Programs

Maple

seq(numtheory:-order(10, 2^ithprime(i)-1),i=1..20); # Robert Israel, May 25 2020

Mathematica

f[n_] := Block[{ds = Divisors[n - 1]}, p = Position[ PowerMod[10, ds, n], 1]; If[p == {}, Length[ RealDigits[1/n][[1, 1]]], Take[ds, p[[1, 1]]][[ -1]]]]; Table[ f[2^Prime[n] - 1], {n, 11}]

Extensions

More terms from Robert Israel, May 25 2020