1,1

Primes p such that the decimal expansion of 1/p has period p-1.

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

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.

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.

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

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.

L. J. Goldstein, Density questions in algebraic number theory, Amer. Math. Monthly, 78 (1971), 342-349.

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 Th\'{e}orie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 61.

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

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

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].

B. Chanco, Full Reptend Prime

Pieter Moree, Artin's primitive root conjecture - a survey

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

D. Williams, Primitive Roots (Check)

Index entries for primes by primitive root

Index entries for sequences related to decimal expansion of 1/n

f[n_]:=Block[{q}, q=Last[First[RealDigits[1/n]]]; If[IntegerQ[q], q={}]; FromDigits[q]]; q=0; lst={}; Do[If[StringLength[ToString[f[n]]]>q, AppendTo[lst, n]; q=StringLength[ToString[f[n]]]], {n, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), May 21 2009]

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

Apart from initial term, identical to A006883.

Other definitions of cyclic numbers: A003277, A001914. Cf. A005596, A001122, A048296.

Sequence in context: A101240 A058887 A167797 this_sequence A071845 A084704 A156005

Adjacent sequences: A001910 A001911 A001912 this_sequence A001914 A001915 A001916

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).