A cyclic number is an -digit integer that, when multiplied by 1, 2, 3, ..., , produces the same digits in a different order. Cyclic numbers are generated by the full reptend primes, i.e., 7, 17, 19, 23, 29, 47, 59, 61, 97, ... (OEIS A001913).
The numbers of cyclic numbers for , 1, 2, ... are 0, 1, 9, 60, 467, 3617, 25883, 248881, 2165288, 19016617, 170169241, ... (OEIS A086018). It has been conjectured, but not yet proven, that an infinite number of cyclic numbers exist. In fact, the fraction of cyclic numbers out of all primes has been conjectured to be Artin's constant. The fraction of cyclic numbers among primes is 0.3739551.
When a cyclic number is multiplied by its generator, the result is a string of 9s. This is a special case of Midy's theorem.
See Yates (1973) for a table of prime period lengths for primes.
-digit integer that, when multiplied by 1, 2, 3, ..., 
, produces the same digits in a different order. Cyclic numbers are generated by the full reptend primes, i.e., 7, 17, 19, 23, 29, 47, 59, 61, 97, ... (OEIS A001913). 
for 
, 1, 2, ... are 0, 1, 9, 60, 467, 3617, 25883, 248881, 2165288, 19016617, 170169241, ... (OEIS A086018). It has been conjectured, but not yet proven, that an infinite number of cyclic numbers exist. In fact, the fraction of cyclic numbers out of all primes has been conjectured to be Artin's constant 
. The fraction of cyclic numbers among primes 
is 0.3739551. 
. 
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