A001913 - OEIS

%I M4353 N1823 #124 Feb 16 2025 08:32:24

%S 7,17,19,23,29,47,59,61,97,109,113,131,149,167,179,181,193,223,229,

%T 233,257,263,269,313,337,367,379,383,389,419,433,461,487,491,499,503,

%U 509,541,571,577,593,619,647,659,701,709,727,743,811,821,823,857,863,887,937,941,953,971,977,983

%N Full reptend primes: primes with primitive root 10.

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

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

%C 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.

%C 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.

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

%C 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

%C The number of terms < 10^n: A086018(n). - _Robert G. Wilson v_, Aug 18 2014

%D 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.

%D Albert H. Beiler, Recreations in the Theory of Numbers, 2nd ed. New York: Dover, 1966, pages 65, 309.

%D John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 161.

%D C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 380.

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 115.

%D M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 61.

%D H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), Ch. 19, 'Die periodischen Dezimalbrüche'.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H Robert Israel, <a href="/A001913/b001913.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H B. Chanco, <a href="http://bchanco.free.fr/frp/ArtinIntro.html">Full Reptend Prime</a>

%H Sebastian M. Cioabă and Werner Linde, <a href="https://bookstore.ams.org/view?ProductCode=AMSTEXT/58">A Bridge to Advanced Mathematics: from Natural to Complex Numbers</a>, Amer. Math. Soc. (2023) Vol. 58, see page 186.

%H L. J. Goldstein, <a href="http://www.jstor.org/stable/2316895">Density questions in algebraic number theory</a>, Amer. Math. Monthly, 78 (1971), 342-349.

%H Pieter Moree, <a href="http://turing.wins.uva.nl/~moree/varia.htm">Artin's primitive root conjecture - a survey</a>

%H Katsuya Mori, <a href="http://math.colgate.edu/~integers/u77/u77.pdf">On a Certain Inverse Problem for Carousel Numbers</a>, INTEGERS 20 (2020), #A77.

%H OEIS Wiki, <a href="/wiki/Full_reptend_primes">Full reptend primes</a>

%H Matt Parker and Brady Haran, <a href="https://www.youtube.com/watch?v=DmfxIhmGPP4">The Reciprocals of Primes</a>, Numberphile video (2022)

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclicNumber.html">Cyclic Number.</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DecimalExpansion.html">Decimal Expansion.</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FullReptendPrime.html">Full Reptend Prime.</a>

%H D. Williams, <a href="http://www.louisville.edu/~dawill03/crypto/Primitive.html">Primitive Roots (Check)</a> [Dead link]

%H Chai Wah Wu, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.121.06.529">Pigeonholes and repunits</a>, Amer. Math. Monthly, 121 (2014), 529-533.

%H <a href="/index/Pri#primes_root">Index entries for primes by primitive root</a>

%H <a href="/index/1#1overn">Index entries for sequences related to decimal expansion of 1/n</a>

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

%p A001913 := proc(n) local st, period:

%p st := ithprime(n):

%p period := numtheory[order](10,st):

%p if (st-1 = period) then

%p    RETURN(st):

%p fi: end:  seq(A001913(n), n=1..200); # _Jani Melik_, Feb 25 2011

%t pr=10; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

%t (* Second program: *)

%t Join[{7},Select[Prime[Range[300]],PrimitiveRoot[#,10]==10&]] (* _Harvey P. Dale_, Feb 01 2018 *)

%o (PARI) forprime(p=7,1e3,if(znorder(Mod(10,p))+1==p,print1(p", "))) \\ _Charles R Greathouse IV_, Feb 27 2011

%o (PARI) is(n)=Mod(10,n)^(n\2)==-1 && isprime(n) && znorder(Mod(10,n))+1==n \\ _Charles R Greathouse IV_, Oct 24 2013

%Y Apart from initial term, identical to A006883.

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

%Y Cf. A005596, A001122, A048296, A051626.

%K nonn,easy,nice

%O 1,1

%A _N. J. A. Sloane_