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%I #9 May 26 2020 13:49:12
%S 1,6,15,42,44,1365,3855,74898,44620,39672,195225786,616318176,
%T 26815350376,1186422030,663226400,1001874900,4885233465012400,
%U 1152921504606846975,10197205073773110,9758202933231640,2908370863958880,1152589154156603558,2895630705663782454386,103161669940448356241593685
%N Periodicity of the reciprocal of the Mersenne numbers (A001348).
%C "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]
%D Keith Ball, Strange Curves, Counting Rabbits and other Mathematical Explorations, Princeton University Press, Princeton and Oxford, 2003, Page 57.
%p seq(numtheory:-order(10, 2^ithprime(i)-1),i=1..20); # _Robert Israel_, May 25 2020
%t 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}]
%Y Cf. A001348.
%K base,nonn
%O 1,2
%A _Robert G. Wilson v_, Nov 22 2004
%E More terms from _Robert Israel_, May 25 2020
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