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# Full reptend primes

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A full reptend prime (long period prime or long prime, maximal period prime) in base
 b
is a prime number
 p
such that the formula
 b  p  − 1 − 1 p
, pb,
where
 p
does not divide
 b
, gives a cyclic number (with
 p  −  1
digits). Therefore the digital expansion of
 1 p
in base
 b
repeats the digits of the corresponding cyclic number infinitely. (Base 10 is assumed if no base is specified.)

## Examples

A020806: Decimal expansion of
 1 7
.
0.14285714285714...
7 is a “full reptend prime” in base 10 since
 1 7
has period length 6. Since
 10 6  −  1 7
=
 999999 7
= 142857
is a cyclic number, we have
 n
 142857  ×  n
 n 7
1 142857 × 1 = 142857 0.142857142857...
2 142857 × 2 = 285714 0.285714285714...
3 142857 × 3 = 428571 0.428571428571...
4 142857 × 4 = 571428 0.571428571428...
5 142857 × 5 = 714285 0.714285714285...
6 142857 × 6 = 857142 0.857142857142...

## Sequences

A001913 Full reptend primes: primes with primitive root 10.

{7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, ...}
A180340 Numbers with
 x
digits such that the first
 x
multiples are cyclic permutations of the number, leading 0’s omitted (or cyclic numbers). (Periods of reciprocals of A001913.)
{142857, 5882352941176470, 526315789473684210, 4347826086956521739130, 3448275862068965517241379310, 2127659574468085106382978723404255319148936170, 1694915254237288135593220338983050847457627118644067796610, ...}
A002371 Period of decimal expansion of
 1 / (n-th prime)
(0 by convention for the primes 2 and 5).
{0, 1, 0, 6, 2, 6, 16, 18, 22, 28, 15, 3, 5, 21, 46, 13, 58, 60, 33, 35, 8, 13, 41, 44, 96, 4, 34, 53, 108, 112, 42, 130, 8, 46, 148, 75, 78, 81, 166, 43, 178, 180, 95, 192, 98, 99, ...}
A006883 Long period primes: the decimal expansion of
 1 p
has period
 p  −  1
. (2 divides 10, thus should not have been considered, see A001913.)
{2, 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, ...}

A004042 Periods of reciprocals of A006883, starting with first nonzero digit. (2 divides 10, thus should not have been considered, see A001913.)

{0, 142857, 5882352941176470, 526315789473684210, 4347826086956521739130, 3448275862068965517241379310, 2127659574468085106382978723404255319148936170, 1694915254237288135593220338983050847457627118644067796610, ...}