login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A006883 Long period primes: the decimal expansion of 1/p has period p-1.
(Formerly M1745)
20
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.

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)...

-------------------------------------------------

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

(End)

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. A006559, A067556.

Sequence in context: A073998 A129444 A079815 * A023269 A023300 A045378

Adjacent sequences:  A006880 A006881 A006882 * A006884 A006885 A006886

KEYWORD

nonn,nice,easy,base

AUTHOR

Robert Munafo

EXTENSIONS

More terms from James A. Sellers, Aug 21 2000

Additional comments from Jason Earls (zevi_35711(AT)yahoo.com), Apr 06 2001

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 26 23:19 EDT 2017. Contains 284141 sequences.