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, 571, 577, 593, 619, 647, 659, 701, 709, 727, 743, 811, 821, 823, 857, 863, 887, 937, 941, 953, 971, 977, 983

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Primes p such that the decimal expansion of 1/p has period p-1, which is the greatest period possible for any integer.

Primes p such that the corresponding entry in A002371 is p-1.

Pieter Moree writes (Oct 20 2004): Assuming the Generalized Riemann Hypothesis it can be shown that the density of primes p such that a prescribed integer g has order (p-1)/t, with t fixed exists and, moreover, it can be computed. This density will be a rational number times the so-called Artin constant. For 2 and 10 the density of primitive roots is A, the Artin constant itself.

R. K. Guy writes (Oct 20 2004): MR 2004j:11141 speaks of the unearthing by Lenstra & Stevenhagen of correspondence concerning the density of this sequence between the Lehmers & Artin.

Also called long period primes, long primes or maximal period primes.

The base-10 cyclic numbers A180340, (b^(p-1) - 1) / p, with b = 10, are obtained from the full reptend primes p. - Daniel Forgues, Dec 17 2012

The number of terms < 10^n: A086018(n). - Robert G. Wilson v, Aug 18 2014

Examples

			7 is in the sequence because 1/7 = 0.142857142857... and the period = 7-1 = 6.
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 864.
• Albert H. Beiler, Recreations in the Theory of Numbers, 2nd ed. New York: Dover, 1966, pages 65, 309.
• John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 161.
• C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 380.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 115.
• M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 61.
• H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), Ch. 19, 'Die periodischen Dezimalbrüche'.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Robert Israel, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• B. Chanco, Full Reptend Prime
• Sebastian M. Cioabă and Werner Linde, A Bridge to Advanced Mathematics: from Natural to Complex Numbers, Amer. Math. Soc. (2023) Vol. 58, see page 186.
• L. J. Goldstein, Density questions in algebraic number theory, Amer. Math. Monthly, 78 (1971), 342-349.
• Pieter Moree, Artin's primitive root conjecture - a survey
• Katsuya Mori, On a Certain Inverse Problem for Carousel Numbers, INTEGERS 20 (2020), #A77.
• OEIS Wiki, Full reptend primes
• Matt Parker and Brady Haran, The Reciprocals of Primes, Numberphile video (2022)
• Eric Weisstein's World of Mathematics, Cyclic Number.
• Eric Weisstein's World of Mathematics, Decimal Expansion.
• Eric Weisstein's World of Mathematics, Full Reptend Prime.
• D. Williams, Primitive Roots (Check) [Dead link]
• Chai Wah Wu, Pigeonholes and repunits, Amer. Math. Monthly, 121 (2014), 529-533.
• Index entries for primes by primitive root
• Index entries for sequences related to decimal expansion of 1/n

Crossrefs

Apart from initial term, identical to A006883.

Other definitions of cyclic numbers: A003277, A001914, A180340.

Cf. A005596, A001122, A048296, A051626.

Programs

Maple

A001913 := proc(n) local st, period:
st := ithprime(n):
period := numtheory[order](10,st):
if (st-1 = period) then
   RETURN(st):
fi: end:  seq(A001913(n), n=1..200); # Jani Melik, Feb 25 2011

Mathematica

pr=10; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]
(* Second program: *)
Join[{7},Select[Prime[Range[300]],PrimitiveRoot[#,10]==10&]] (* Harvey P. Dale, Feb 01 2018 *)

PARI

forprime(p=7,1e3,if(znorder(Mod(10,p))+1==p,print1(p", "))) \\ Charles R Greathouse IV, Feb 27 2011

PARI

is(n)=Mod(10,n)^(n\2)==-1 && isprime(n) && znorder(Mod(10,n))+1==n \\ Charles R Greathouse IV, Oct 24 2013

Python

from itertools import count, islice
from sympy import nextprime, n_order
def A001913_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)==p-1:
            yield p
A001913_list = list(islice(A001913_gen(),20)) # Chai Wah Wu, Mar 03 2025