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

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, 541, 571, 577, 593, 619, 647, 659, 701, 709, 727, 743, 811, 821, 823, 857, 863

Offset: 1

Author

Robert Munafo

Keywords

• nonn
• nice
• easy
• base

Comments

Also called full reptend primes or maximal period primes.

Also called golden primes or long primes.

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

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.
• Carl Friedrich Gauss, "Disquisitiones Arithmeticae"
• 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.
• 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.]
• C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, 1966, pp. 56-58.
• 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
• 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].
• Eric Weisstein's World of Mathematics, Full Reptend Prime.
• Index entries for sequences related to decimal expansion of 1/n

Crossrefs

Apart from initial term, identical to A001913.

Cf. A005596, A006559, A067556.

Cf. A001122 (long period primes in binary).

Programs

Maple

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

Mathematica

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 *)
maxPeriodQ[p_] := MultiplicativeOrder[10, p] == p-1; maxPeriodQ[2] = True; Select[ Prime[ Range[150]], maxPeriodQ] (* Jean-François Alcover, Jan 07 2013 *)

PARI

print1(2);forprime(p=7,1e3,if(znorder(Mod(10,p))+1==p,print1(", "p))) \\ Charles R Greathouse IV, Feb 27 2011

Formula

From Gerard Schildberger, Jul 02 2005: (Start)

Emil Artin conjectured that the proportion of primes that belong to this sequence can be expressed as:

(2*1-1)(3*2-1)(5*4-1)(7*6-1)(11*10-1)(13*12-1)...

------------------------------------------------- = 0.373955813619202288...

(2*1)(3*2)(5*4)(7*6)(11*10)(13*12)...

(End)

This Artin's constant, Product_{p prime} (1-1/(p^2-p)), is referenced in A005596. - Robert FERREOL, Jun 05 2018

Extensions

More terms from James Sellers, Aug 21 2000

Additional comments from Jason Earls, Apr 06 2001