A005596 -id:A005596 - cp's OEIS frontend

A374829 Decimal expansion of 5(5 - 3A136141)/36.

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

Offset: 0

Stefano Spezia, Jul 21 2024

Lower bound of Artin's constant A005596.

			0.3722958323403631411676246695044684462699688877431169243...

Rafael Jakimczuk, Numbers of the form p-1 where p is prime, July 2024. See pp. 16-17.

Cf. A005596, A136141, A374830.

5*(5 - 3*sumeulerrat(1/(p*(p-1))))/36 \\ Amiram Eldar, Aug 20 2024

Equals (1/2)*(5/6)*(1 - (A136141 - (1/2 + 1/6))) (see Jakimczuk).

Equals (153/125)*A374830.

A374831 Decimal expansion of Product_{p prime} (1 - (1/(p(p - 1)))p^2/(p^2 + 1)).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jul 21 2024

			0.4589374985054359613063426181...

Rafael Jakimczuk, Numbers of the form p-1 where p is prime, July 2024. See p. 23.

Cf. A005596, A005597, A065414, A065418, A065419, A374830 (lower bound).

PARI
```
prodeulerrat(1-p^2/(p*(p-1)*(p^2+1)))
```

A098937 Number of cyclic numbers, primes with primitive root 10, (A001913) in the first 10^n primes (A000040).

Original entry on oeis.org

5, 38, 387, 3755, 37523, 374126, 3740610, 37393725, 373953691, 3739544360

Offset: 1

Robert G. Wilson v, Oct 19 2004

Cf. A001913, A006883, A000040, A005596.

f[n_Integer] := Block[{ds = Divisors[n - 1]}, (n - 1)/Take[ds, Position[ PowerMod[10, ds, n], 1][[1, 1]]][[ -1]]]; c = 0; k = 4; Do[ While[k <= 10^n, a = f[ Prime[k]]; If[a == 1, c++ ]; k++ ]; Print[c], {n, 7}]

Lim_{n->oo} a(n)/10^n = Artin's constant (A005596).

a(8)-a(10) from Amiram Eldar, Jul 04 2021

A119964 Numerator of the n-th Artin product.

Original entry on oeis.org

1, 5, 19, 779, 84911, 2632241, 713337311, 1163866139, 587752400195, 476667196558145, 2856927907113011, 345688276760674331, 13819099649042566549, 4988694973304366524189, 10780569837310736058772429

Offset: 1

Alexander Adamchuk, Aug 03 2006

Artin's constant (A005596) is equal to Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,Infinity}]. n-th Artin product is Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}].

Eric Weisstein's World of Mathematics, Artin's Constant.

Cf. A119534, A091568, A091567, A005596, A048296.

Table[Numerator[Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}]],{n,1,20}]

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

A119978 Denominator of the n-th Artin product.

Original entry on oeis.org

2, 12, 48, 2016, 221760, 6918912, 1881944064, 3079544832, 1558249684992, 1265298744213504, 7591792465281024, 919297051250393088, 36771882050015723520, 13282003796465679335424

Offset: 1

Alexander Adamchuk, Aug 03 2006

Artin's constant (A005596) is equal to Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,Infinity}]. n-th Artin product is Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}].

Eric Weisstein's World of Mathematics, Artin's Constant.

Cf. A119534, A091568, A091567, A005596, A048296.

Table[Denominator[Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}]],{n,1,20}]

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

A271877 Decimal expansion of Matthews' constant C_4, an analog of Artin's constant for primitive roots.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 16 2016

			0.026107446314917708083...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.4 Artin's constant, p. 105.

K. R. Matthews, A generalisation of Artin's conjecture for primitive roots, Acta arithmetica, Vol. 29, No. 2 (1976), pp. 113-146.

Cf. A005596, A065414, A065415, A065416, A271780, A271798, A271869.

$MaxExtraPrecision = 2000; LR = LinearRecurrence[{2, 3, -10, 10, -5, 1}, {0, -8, 6, -40, 35, -194}, 10^4]; r[n_Integer] := LR[[n]]; NSum[r[n] PrimeZetaP[n]/n, {n, 2, Infinity}, NSumTerms -> 2000, WorkingPrecision -> 300, Method -> "AlternatingSigns"] // Exp // RealDigits[#, 10, 20]& // First // Prepend[#, 0]&
$MaxExtraPrecision = 1000; Clear[f]; f[p_] := 1 - (p^4 - (p - 1)^4)/(p^4*(p - 1)); Do[c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; Print[f[2] * Exp[N[Sum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 105]]], {m, 100, 1000, 100}] (* Vaclav Kotesovec, Jun 19 2020 *)

prodeulerrat(1 - (p^4 - (p - 1)^4)/(p^4*(p - 1))) \\ Amiram Eldar, Mar 16 2021

C_4 = Product_{p prime} 1 - (p^4 - (p - 1)^4)/(p^4*(p - 1)).

More digits from Vaclav Kotesovec, Jun 19 2020

A340565 Decimal expansion of the Product_{lesser twin primes p == 5 (mod 6)} 1/(1 - 1/p^2).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 11 2021

Lesser twin primes A001359 (with the exception of the first prime, 3) are congruent to 5 mod 6: this constant is smaller than A340576.
By extrapolating method most probably the next two decimal digits are 1.056932291(46).
The known high-precision algorithms for Euler products are based on the Dirichlet L function and the Moebius inversion formula (see Mathematica procedure of Jean-François Alcover in A175646).
The constant is between 1.056932291453... and 1.056932291494. - R. J. Mathar, Feb 14 2025

			1.0569322914...

Cf. A005596, A065463, A065465, A065483, A065485, A065489, A065493, A072691, A082695, A111003, A175646.

One more digit confirmed by a bracketing of partial products - R. J. Mathar, Feb 14 2025

A373774 a(n) is the number of terms of A000057 in [10^n].

Original entry on oeis.org

3, 7, 38, 249, 1894, 15456, 130824, 11344404, 10007875, 89562047

Offset: 1

Stefano Spezia, Jun 18 2024

Rishi Kumar, Kepler Sets of Second-Order Linear Recurrence Sequences Over Q_p, arXiv:2406.05890 [math.NT], 2024. See pp. 6-7.

Cf. A000045, A000057, A001602, A005596, A006880.

a[n_]:=Length[Select[Prime[Range[PrimePi[10^n]]], Function[p, c=0; b=1; m=1; While[b != 0, t=b; b = Mod[(c+b), p]; c=t; m++]; m>p]]]; Array[a,4] (* after Charles R Greathouse IV and Jean-François Alcover in A000057 *)

Conjectured by Kumar under GRH: Limit_{n->oo} a(n)/A006880(n) = (10/19)*A005596 = 0.196818849273264362134067397024429692164015394...

A387331 Least k such that n*k + 1 is a prime > 2 with 2 as a primitive root; a(n) = 0 if no such k exists.

Original entry on oeis.org

2, 1, 4, 1, 2, 2, 4, 0, 2, 1, 6, 1, 4, 2, 4, 0, 26, 1, 22, 3, 10, 3, 6, 0, 4, 2, 6, 1, 2, 2, 12, 0, 2, 13, 6, 1, 4, 11, 14, 0, 2, 5, 4, 15, 4, 3, 14, 0, 4, 2, 12, 1, 2, 3, 12, 0, 26, 1, 12, 1, 58, 6, 6, 0, 2, 1, 4, 9, 2, 3, 12, 0, 4, 2, 38, 25, 50, 7, 4, 0, 2

Offset: 1

Pablo Cadena-Urzúa, Aug 26 2025

If n is a multiple of 8 then a(n) = 0.
Proof: If n = 8*m, then any prime p = n*k+1 satisfies p == 1 (mod 8). Thus 2 is a quadratic residue modulo p, so ord_p(2) | (p-1)/2 and cannot equal p-1.
Computations show that for 1 <= n <= 2000 with 8 !| n, a(n) > 0. This agrees with Artin's conjecture: for each n with 8 !| n there should be infinitely many primes p == 1 (mod n) with 2 as a primitive root.

			a(1) = 2 since 1*2+1 = 3 is prime and 2 generates (Z/3Z)^*.
a(2) = 1 since 2*1+1 = 3 is prime and 2 is a primitive root modulo 3.
a(3) = 4 since 3*4+1 = 13 is prime and ord_13(2) = 12.
a(8) = 0 because every 8*k+1 == 1 (mod 8).

E. Artin, Collected Papers, Addison-Wesley, 1965.

Pablo Cadena-Urzua, Table of n, a(n) for n = 1..2000

Cf. A001122, A005596.

a := proc(n) local k, p;
  if n mod 8 = 0 then return 0 fi;
  for k from 1 do
    p := n*k+1;
    if isprime(p) and p>2 then
      if order(Mod(2, p)) = p-1 then return k fi;
    fi;
  od;
end;

a[n_] := If[Mod[n, 8]==0, 0, Module[{k=1, p, fac}, While[True,
  p=n*k+1;
  If[p>2 && PrimeQ[p], fac=FactorInteger[p-1][[All, 1]];
  If[And@@(PowerMod[2, (p-1)/#, p]!=1&/@fac), Return[k]]];
  k++]]]

from sympy import isprime, factorint
def is_primitive_root_base2(p):
    phi = p-1
    return all(pow(2, phi//q, p)!=1 for q in factorint(phi))
def a(n, kmax=10**7):
    if n%8==0: return 0
    for k in range(1, kmax+1):
        p = n*k+1
        if p>2 and isprime(p) and is_primitive_root_base2(p):
            return k
    return 0

a(n) = 0 if n == 0 (mod 8).

Otherwise, a(n) = min { k >= 1 : p = n*k+1 is prime > 2 and ord_p(2) = p-1 }.

cp's OEIS Frontend

A374829 Decimal expansion of 5*(5 - 3*A136141)/36.

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A374831 Decimal expansion of Product_{p prime} (1 - (1/(p*(p - 1)))*p^2/(p^2 + 1)).

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A098937 Number of cyclic numbers, primes with primitive root 10, (A001913) in the first 10^n primes (A000040).

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A119964 Numerator of the n-th Artin product.

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A119978 Denominator of the n-th Artin product.

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A271877 Decimal expansion of Matthews' constant C_4, an analog of Artin's constant for primitive roots.

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A340565 Decimal expansion of the Product_{lesser twin primes p == 5 (mod 6)} 1/(1 - 1/p^2).

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A373774 a(n) is the number of terms of A000057 in [10^n].

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A374829 Decimal expansion of 5(5 - 3A136141)/36.

A374831 Decimal expansion of Product_{p prime} (1 - (1/(p(p - 1)))p^2/(p^2 + 1)).