cp's OEIS Frontend

A006883 Long period primes: the decimal expansion of 1/p has period p-1.

%I A006883 M1745 #77 Jul 02 2025 16:01:55
%S A006883 2,7,17,19,23,29,47,59,61,97,109,113,131,149,167,179,181,193,223,229,
%T A006883 233,257,263,269,313,337,367,379,383,389,419,433,461,487,491,499,503,
%U A006883 509,541,571,577,593,619,647,659,701,709,727,743,811,821,823,857,863
%N A006883 Long period primes: the decimal expansion of 1/p has period p-1.
%C A006883 Also called full reptend primes or maximal period primes.
%C A006883 Also called golden primes or long primes.
%C A006883 Here, as opposed to A001913, 2 is a term, because the decimal expansion of 1/2 is 0.5000000000..., so it is periodic with period 1 and pattern 0. - _Michel Marcus_, Jun 06 2018
%D A006883 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 A006883 Albert H. Beiler, Recreations in the Theory of Numbers, 2nd ed. New York: Dover, 1966, pages 65, 309.
%D A006883 John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 161.
%D A006883 Carl Friedrich Gauss, "Disquisitiones Arithmeticae"
%D A006883 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 115.
%D A006883 M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 61.
%D A006883 D. H. Lehmer, A note on primitive roots, Scripta Mathematica, vol. 26 (1963), p. 117. [Gives some interesting information about the frequency of maximal period primes and discusses two freak cases.]
%D A006883 C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, 1966, pp. 56-58.
%D A006883 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006883 T. D. Noe, <a href="/A006883/b006883.txt">Table of n, a(n) for n=1..1000</a>
%H A006883 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 A006883 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FullReptendPrime.html">Full Reptend Prime.</a>
%H A006883 <a href="/index/1#1overn">Index entries for sequences related to decimal expansion of 1/n</a>
%F A006883 From _Gerard Schildberger_, Jul 02 2005: (Start)
%F A006883 Emil Artin conjectured that the proportion of primes that belong to this sequence can be expressed as:
%F A006883 (2*1-1)(3*2-1)(5*4-1)(7*6-1)(11*10-1)(13*12-1)...
%F A006883 ------------------------------------------------- = 0.373955813619202288...
%F A006883 (2*1)(3*2)(5*4)(7*6)(11*10)(13*12)...
%F A006883 (End)
%F A006883 This Artin's constant, Product_{p prime} (1-1/(p^2-p)), is referenced in A005596. - _Robert FERREOL_, Jun 05 2018
%p A006883 isA006883 := proc(p) if p = 2 then true; elif isprime(p) then RETURN( numtheory[order](10,p) = p-1) ; else false; fi; end: for i from 1 to 300 do p := ithprime(i) ; if isA006883(p) then printf("%d ",p) ; fi; od: # _R. J. Mathar_, Apr 01 2009
%t A006883 f[n_Integer] := Block[{ds = Divisors[n - 1]}, (n - 1)/Take[ ds, Position[ PowerMod[ 10, ds, n], 1] [[1, 1]]] [[ -1]]]; Select[ Prime[ Range[4, 150]], f[ # ] == 1 &] (* _Robert G. Wilson v_, Sep 14 2004 *)
%t A006883 maxPeriodQ[p_] := MultiplicativeOrder[10, p] == p-1; maxPeriodQ[2] = True; Select[ Prime[ Range[150]], maxPeriodQ] (* _Jean-François Alcover_, Jan 07 2013 *)
%o A006883 (PARI) print1(2);forprime(p=7,1e3,if(znorder(Mod(10,p))+1==p,print1(", "p))) \\ _Charles R Greathouse IV_, Feb 27 2011
%Y A006883 Apart from initial term, identical to A001913.
%Y A006883 Cf. A005596, A006559, A067556.
%Y A006883 Cf. A001122 (long period primes in binary).
%K A006883 nonn,nice,easy,base
%O A006883 1,1
%A A006883 _Robert Munafo_
%E A006883 More terms from _James Sellers_, Aug 21 2000
%E A006883 Additional comments from _Jason Earls_, Apr 06 2001