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A006883 Long period primes: the decimal expansion of 1/p has period p-1.
(Formerly M1745)
21
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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, "Disquitiones 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

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

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 *)

PROG

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

CROSSREFS

Apart from initial term, identical to A001913.

Cf. A005596, A006559, A067556.

Cf. A001122 (long period primes in binary).

Sequence in context: A073998 A129444 A079815 * A023269 A023300 A045378

Adjacent sequences:  A006880 A006881 A006882 * A006884 A006885 A006886

KEYWORD

nonn,nice,easy,base,changed

AUTHOR

Robert Munafo

EXTENSIONS

More terms from James A. Sellers, Aug 21 2000

Additional comments from Jason Earls, Apr 06 2001

STATUS

approved

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Last modified June 19 08:41 EDT 2018. Contains 305581 sequences. (Running on oeis4.)