A065414 -id:A065414 - cp's OEIS frontend

A146485 Decimal expansion of Product_{n>=2} (1 - 1/(n^2*(n-1))).

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

Offset: 0

R. J. Mathar, Feb 13 2009

Product of Artin's constant of rank 2 and the equivalent almost-prime products.

			0.6739173633763... = (1 - 1/4)*(1 - 1/18)*(1 - 1/48)*(1 - 1/100)*...

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], table 3, first line with r=2.

Cf. A065414.

r := 2 : ni := fsolve( (n+1)^r*n-1,n,complex) : 1.0/mul(GAMMA(1-d),d=ni) ; # R. J. Mathar, Feb 20 2009

g[k_] := Gamma[Root[-3 + 8# - 5#^2 + #^3 & , k]]; RealDigits[1/(g[1]*g[2]*g[3]) // Re, 10, 105] // First (* Jean-François Alcover, Feb 12 2013 *)

The logarithm is -Sum_{s>=2} Sum_{j=1..floor(s/(1+r))} binomial(s-r*j-1, j-1)*(1-Zeta(s))/j at r = 2.
s*Sum_{j=1..floor(s/3)} binomial(s-2j-1, j-1)/j = A001609(s)-1.
Equals 1/Product_{k=1..3} Gamma(1-x_k), where x_k are the 3 roots of the polynomial x*(x+1)^2-1. [R. J. Mathar, Feb 20 2009]

More terms from Jean-François Alcover, Feb 12 2013

A146486 Decimal expansion of Product_{q in A001358} (1-1/(q^2*(q-1))).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 13 2009

Semiprime analog of A065414.

			0.969932325001525316214920... = (1-1/48)*(1-1/180)*(1-1/648)*(1-1/900)*..

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products.., arxiv:0903.2514, variable A_2^(2), Table 3.

A146487 Decimal expansion of Product_{q in A014612} (1-1/(q^2*(q-1))).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 13 2009

3-almost prime analog of A065414.

			0.99659892748024127.. = (1-1/448)*(1-1/1584)*(1-1/5508)*(1-1/7600)*..

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], table 3 at r=2, k=3. [From _R. J. Mathar_, Mar 28 2009]

The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_3(s)/j at r=2, where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.

A146488 Decimal expansion of Product_{q in A014613} (1-1/(q^2*(q-1))).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 13 2009

4-almost prime analog of A065414.

			0.99959527858653553563... = (1-1/3840)*(1-1/13248)*(1-1/45360)*(1-1/62400)*..

The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_4(s)/j at r=2, where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.

A271869 Decimal expansion of Matthews' constant C_3, an analog of Artin's constant for primitive roots.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 16 2016

			0.0608216551203050860056322754619208554313373734757679419826434...

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.

digits = 70; $MaxExtraPrecision = 1000; m0 = 2000; dm = 200; Clear[s]; LR =
LinearRecurrence[{2, 2, -6, 4, -1}, {0, 6, 0, 22, 5}, 2 m0]; r[n_Integer] := LR[[n]]; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 2, m}, NSumTerms -> 2 m0, WorkingPrecision -> digits+10] // Exp; s[m0]; s[m = m0+dm]; While[RealDigits[s[m], 10, digits][[1]] != RealDigits[ s[m-dm], 10, digits][[1]], Print[m]; m = m + dm]; Join[{0}, RealDigits[ s[m], 10, digits][[1]]]

prodeulerrat(1 - (p^3 - (p - 1)^3)/(p^3*(p - 1))) \\ Amiram Eldar, Mar 16 2021

C_3 = Product_{p prime} 1 - (p^3 - (p - 1)^3)/(p^3*(p - 1)).

More digits from Vaclav Kotesovec, Jun 19 2020

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)))
```

A066517 Continued fraction expansion of Artin constant of rank 2: product(1-1/(p^3-p^2), p=prime).

Original entry on oeis.org

0, 1, 2, 3, 3, 1, 2, 2, 1, 3, 1, 8, 1, 4, 1, 1, 2, 1, 2, 31, 9, 2, 1, 3, 1, 3, 13, 3, 2, 11, 3, 1, 1, 1, 2, 2, 10, 10, 2, 1, 204, 4, 2, 12, 2, 8, 1, 1, 6, 17, 5, 2, 34, 4, 2, 2, 1, 5, 1, 1, 1, 1, 4, 1, 2, 1, 54, 3, 1, 6, 1, 13, 3, 2, 12, 1, 1, 1, 2, 2, 5, 2, 2, 7, 2, 2, 2, 1, 2, 1, 10, 3, 3, 1, 8, 9, 1

Offset: 0

Simon Plouffe Jan 05 2002

nonn

			0.697501358496365903284670350820922924...

G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
Index entries for sequences related to Artin's conjecture

Cf. A065414.

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

cp's OEIS Frontend

A146485 Decimal expansion of Product_{n>=2} (1 - 1/(n^2*(n-1))).

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A146486 Decimal expansion of Product_{q in A001358} (1-1/(q^2*(q-1))).

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A146487 Decimal expansion of Product_{q in A014612} (1-1/(q^2*(q-1))).

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A146488 Decimal expansion of Product_{q in A014613} (1-1/(q^2*(q-1))).

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A271869 Decimal expansion of Matthews' constant C_3, an analog of Artin's constant for primitive roots.

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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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A066517 Continued fraction expansion of Artin constant of rank 2: product(1-1/(p^3-p^2), p=prime).

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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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Programs

Mathematica

PARI

Formula

Extensions

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