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A006883
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Long period primes: the decimal expansion of 1/p has period p-1.
(Formerly M1745)
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25
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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].
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI) print1(2); forprime(p=7, 1e3, if(znorder(Mod(10, p))+1==p, print1(", "p))) \\ Charles R Greathouse IV, Feb 27 2011
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CROSSREFS
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Apart from initial term, identical to A001913.
Cf. A001122 (long period primes in binary).
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KEYWORD
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nonn,nice,easy,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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