A006559 Short period primes: the decimal expansion of 1/p has period less than p-1, but greater than zero.

3, 11, 13, 31, 37, 41, 43, 53, 67, 71, 73, 79, 83, 89, 101, 103, 107, 127, 137, 139, 151, 157, 163, 173, 191, 197, 199, 211, 227, 239, 241, 251, 271, 277, 281, 283, 293, 307, 311, 317, 331, 347, 349, 353, 359, 373, 397, 401, 409, 421, 431, 439, 443, 449, 457

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• base

Comments

Primes 2 and 5 are excluded because 1/2 and 1/5 have no period. Also primes p whose multiplicative order mod p is less than p-1.

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Victor Meally, Letter to N. J. A. Sloane, no date.
• Index entries for sequences related to decimal expansion of 1/n

Crossrefs

Cf. A006883.

Programs

Mathematica

Select[Prime[Range[100]], MultiplicativeOrder[10, #] < # - 1 &]

PARI

a(n)=gcd(n,10)==1 && isprime(n) && znorder(Mod(10,n))Charles R Greathouse IV, Mar 15 2014

Python

from itertools import islice
from sympy import nextprime, n_order
def A006559_gen(startvalue=1): # generator of terms >= startvalue
    p = max(startvalue-1,1)
    while (p:=nextprime(p)):
        if p!=2 and p!=5 and n_order(10,p)A006559_list = list(islice(A006559_gen(),20)) # Chai Wah Wu, Mar 03 2025

Extensions

More terms from James Sellers, Aug 21 2000