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A
full reptend prime (
long period prime or
long prime,
maximal period prime) in
base is a
prime number such that the formula
-
where
does not
divide , gives a
cyclic number (with
digits). Therefore the digital expansion of
in base
repeats the digits of the corresponding cyclic number infinitely. (
Base 10 is assumed if no base is specified.)
Examples
A020806: Decimal expansion of
.
-
7 is a “full reptend prime” in base
10 since
has period length
6. Since
is a cyclic number, we have
|
|
|
|
| 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
digits such that the first
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
(
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
has period
.
(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, ...}
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