A048296 -id:A048296 - cp's OEIS frontend

A005596 Decimal expansion of Artin's constant Product_{p=prime} (1-1/(p^2-p)).

3, 7, 3, 9, 5, 5, 8, 1, 3, 6, 1, 9, 2, 0, 2, 2, 8, 8, 0, 5, 4, 7, 2, 8, 0, 5, 4, 3, 4, 6, 4, 1, 6, 4, 1, 5, 1, 1, 1, 6, 2, 9, 2, 4, 8, 6, 0, 6, 1, 5, 0, 0, 4, 2, 0, 9, 4, 7, 4, 2, 8, 0, 2, 4, 1, 7, 3, 5, 0, 1, 8, 2, 0, 4, 0, 0, 2, 8, 0, 8, 2, 3, 4, 4, 3, 0, 4, 3, 1, 7, 0, 8, 7, 2, 5, 0, 5, 6, 8, 9, 8, 1, 6, 0, 3

Offset: 0

N. J. A. Sloane

On Simon Plouffe's web page (and in the book freely available at Gutenberg project) the value is given with an error of +1e-31, as "...651641..." instead of "...641641...". In the reference [Wrench, 1961] cited there, these digits are correct. They are also correct on the Plouffe's Inverter page, as computed by Oliveira e Silva, who comments it took 1 hour at 200 MHz with Mathematica. Using Amiram Eldar's PARI program, the same 500 digits are computed instantly (less than 0.1 sec). - M. F. Hasler, Apr 20 2021

Named after the Austrian mathematician Emil Artin (1898-1962). - Amiram Eldar, Jun 20 2021

			0.37395581361920228805472805434641641511162924860615...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 169.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 0..1000
Ivan Cherednik, A note on Artin's constant, arXiv:0810.2325 [math.NT], 2008.
Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998).
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 156 (constant C7).
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2001; constant A_1^(1).
Pieter Moree, Artin's primitive root conjecture - a survey, arXiv:math/0412262 [math.NT], 2004-2012.
Pieter Moree, The formal series Witt transform, Discr. Math., Vol. 295, No. 1-3 (2005), pp. 143-160. See p. 159.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
G. Niklasch, Artin's constant.
Simon Plouffe, The Artin's Constant=product(1-1/(p**2-p), p=prime) [backup on web.archive.org; chapter 8 of the free Gutenberg.org/ebooks/634]. [Warning: the value given in this reference is incorrect, cf. comment!]
Tomás Oliveira e Silva and Plouffe's Inverter, The first 500 digits of Artin's constant.
Eric Weisstein's World of Mathematics, Artin's constant.
Eric Weisstein's World of Mathematics, Full Reptend Prime.
R. G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993.
John W. Wrench, Jr., Evaluation of Artin's constant and the twin-prime constant, Math. Comp., Vol. 15, No. 76 (1961), pp. 396-398.
Index entries for sequences related to Artin's conjecture.

Cf. A048296, A065414, A001913, A001122.

a = Exp[-NSum[ (LucasL[n] - 1)/n PrimeZetaP[n], {n, 2, Infinity}, PrecisionGoal -> 500, WorkingPrecision -> 500, NSumTerms -> 100000]]; RealDigits[a, 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 taken from Mathematica's Help file on PrimeZetaP *)

prodinf(n=2,1/zeta(n)^(sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1)))/n)) \\ Charles R Greathouse IV, Aug 27 2014

prodeulerrat(1-1/(p^2-p)) \\ Amiram Eldar, Mar 12 2021

Equals Product_{j>=2} 1/Zeta(j)^A006206(j), where Zeta = A013661, A002117 etc. is Riemann's zeta function. - R. J. Mathar, Feb 14 2009
Equals Sum_{k>=1} mu(k)/(k*phi(k)), where mu is the Moebius function (A008683) and phi is the Euler totient function (A000010). - Amiram Eldar, Mar 11 2020
Equals 1/A065488. - Vaclav Kotesovec, Jul 17 2021

More terms from Tomás Oliveira e Silva (http://www.ieeta.pt/~tos)

A001913 Full reptend primes: primes with primitive root 10.

Original entry on oeis.org

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, 887, 937, 941, 953, 971, 977, 983

Offset: 1

N. J. A. Sloane

Primes p such that the decimal expansion of 1/p has period p-1, which is the greatest period possible for any integer.
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.
Also called long period primes, long primes or maximal period primes.
The base-10 cyclic numbers A180340, (b^(p-1) - 1) / p, with b = 10, are obtained from the full reptend primes p. - Daniel Forgues, Dec 17 2012
The number of terms < 10^n: A086018(n). - Robert G. Wilson v, Aug 18 2014

			7 is in the sequence because 1/7 = 0.142857142857... and the period = 7-1 = 6.

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.
C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 380.
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.
H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), Ch. 19, 'Die periodischen Dezimalbrüche'.
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).

Robert Israel, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
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
Sebastian M. Cioabă and Werner Linde, A Bridge to Advanced Mathematics: from Natural to Complex Numbers, Amer. Math. Soc. (2023) Vol. 58, see page 186.
L. J. Goldstein, Density questions in algebraic number theory, Amer. Math. Monthly, 78 (1971), 342-349.
Pieter Moree, Artin's primitive root conjecture - a survey
Katsuya Mori, On a Certain Inverse Problem for Carousel Numbers, INTEGERS 20 (2020), #A77.
OEIS Wiki, Full reptend primes
Matt Parker and Brady Haran, The Reciprocals of Primes, Numberphile video (2022)
Eric Weisstein's World of Mathematics, Cyclic Number.
Eric Weisstein's World of Mathematics, Decimal Expansion.
Eric Weisstein's World of Mathematics, Full Reptend Prime.
D. Williams, Primitive Roots (Check) [Dead link]
Chai Wah Wu, Pigeonholes and repunits, Amer. Math. Monthly, 121 (2014), 529-533.
Index entries for primes by primitive root
Index entries for sequences related to decimal expansion of 1/n

Apart from initial term, identical to A006883.
Other definitions of cyclic numbers: A003277, A001914, A180340.
Cf. A005596, A001122, A048296, A051626.

A001913 := proc(n) local st, period:
st := ithprime(n):
period := numtheory[order](10,st):
if (st-1 = period) then
   RETURN(st):
fi: end:  seq(A001913(n), n=1..200); # Jani Melik, Feb 25 2011

pr=10; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]
(* Second program: *)
Join[{7},Select[Prime[Range[300]],PrimitiveRoot[#,10]==10&]] (* Harvey P. Dale, Feb 01 2018 *)

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

is(n)=Mod(10,n)^(n\2)==-1 && isprime(n) && znorder(Mod(10,n))+1==n \\ Charles R Greathouse IV, Oct 24 2013

from itertools import count, islice
from sympy import nextprime, n_order
def A001913_gen(startvalue=1): # generator of terms >= startvalue
    p = max(startvalue-1,1)
    while (p:=nextprime(p)):
        if p!=2 and p!=5 and n_order(10,p)==p-1:
            yield p
A001913_list = list(islice(A001913_gen(),20)) # Chai Wah Wu, Mar 03 2025

A065488 Decimal expansion of Product_{p prime} (1 + 1/(p^2-p-1)).

Original entry on oeis.org

2, 6, 7, 4, 1, 1, 2, 7, 2, 5, 5, 7, 0, 0, 2, 1, 5, 0, 8, 9, 6, 0, 4, 1, 1, 8, 3, 0, 4, 4, 5, 4, 8, 8, 0, 3, 7, 5, 0, 2, 3, 9, 8, 6, 2, 8, 3, 9, 7, 6, 9, 1, 9, 8, 5, 2, 0, 0, 8, 1, 9, 0, 4, 1, 9, 6, 0, 8, 6, 5, 9, 5, 6, 1, 0, 5, 3, 0, 2, 8, 6, 8, 6, 4, 4, 8, 5, 0, 9, 2, 9, 7, 1, 7, 3, 4, 8, 5, 7

Offset: 1

N. J. A. Sloane, Nov 19 2001

This is 1/Artin's constant, see A005596.

			2.67411272557002150896041183...

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Cf. A005596, A048296, A351219.

$MaxExtraPrecision = 1200; digits = 99; terms = 1200; P[n_] := PrimeZetaP[n ]; LR = Join[{0, 0}, LinearRecurrence[{2, 0, -1}, {2, 3, 6}, term+10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 + 1/(p^2-p-1)) \\ Amiram Eldar, Mar 15 2021

A119534 Largest prime divisor of numerator of the n-th Artin's product.

Original entry on oeis.org

5, 19, 41, 109, 109, 271, 271, 271, 811, 929, 929, 929, 929, 2161, 2161, 2161, 3659, 4421, 4969, 4969, 4969, 4969, 4969, 9311, 10099, 10099, 10099, 10099, 10099, 16001, 17029, 17029, 19181, 22051, 22051, 22051, 22051, 22051, 22051, 22051, 32579

Offset: 2

Alexander Adamchuk, Jul 27 2006

Artin's constant (A005596) is equal to Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,Infinity}]. n-th Artin's product is Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}]. a(n) is prime from A091568 of the form p^2-p-1, where p is prime from A091567.

Amiram Eldar, Table of n, a(n) for n = 2..1000
Eric Weisstein's World of Mathematics, Artin's Constant.

Cf. A091568, A091567, A005596, A048296.

[Max(PrimeDivisors(Numerator(&*[1-1/(NthPrime(k)^2-NthPrime(k)):k in [1..n]]))): n in [2..45]]; // Marius A. Burtea, Feb 18 2020

Table[Max[FactorInteger[Numerator[Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}]]]],{n,2,100}]

a(n) = Max[FactorInteger[Numerator[Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}]]]].

cp's OEIS Frontend

A005596 Decimal expansion of Artin's constant Product_{p=prime} (1-1/(p^2-p)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A001913 Full reptend primes: primes with primitive root 10.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

A065488 Decimal expansion of Product_{p prime} (1 + 1/(p^2-p-1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A119534 Largest prime divisor of numerator of the n-th Artin's product.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A119964 Numerator of the n-th Artin product.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A119978 Denominator of the n-th Artin product.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula