A065416 -id:A065416 - cp's OEIS frontend

A065415 Decimal expansion of Product_{p prime} (1-1/(p^4-p^3)).

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

Offset: 0

N. J. A. Sloane, Nov 15 2001

			0.85654044485354217442616798413595388...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.4, p. 105.

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011, constant A_1^(3) table 3.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Cf. A005596, A065414, A065416.

digits = 99; $MaxExtraPrecision = 400; m0 = 1000; dm = 100; Clear[s]; LR = LinearRecurrence[{2, -1, 0, 1, -1}, {0, 0, 0, 4, 5, 6}, 2 m0]; r[n_Integer] := LR[[n]]; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 3, m}, NSumTerms -> m0, WorkingPrecision -> 400] // 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]; RealDigits[s[m], 10, digits][[1]] (* Jean-François Alcover, Apr 15 2016 *)

prodeulerrat(1-1/(p^4-p^3)) \\ Amiram Eldar, Mar 13 2021

A138407 a(n) = p^4*(p-1), where p = prime(n).

Original entry on oeis.org

16, 162, 2500, 14406, 146410, 342732, 1336336, 2345778, 6156502, 19803868, 27705630, 67469796, 113030440, 143589642, 224465326, 410305012, 702806938, 830750460, 1329973986, 1778817670, 2044673352, 3038106318, 3891582322

Offset: 1

Artur Jasinski, Mar 19 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index to sequences related to prime powers.

Cf. A000010, A030514, A050997, A065416.

[NthPrime((n))^5 - NthPrime((n))^4: n in [1..30] ]; // Vincenzo Librandi, Jun 17 2011

a = {}; Do[p = Prime[n]; AppendTo[a, p^5 - p^4], {n, 1, 50}]; a
f54[n_]:=Module[{c=Prime[n]},c^5-c^4]; Array[f54,30] (* Harvey P. Dale, Mar 29 2015 *)

forprime(p=2,1e3,print1(p^5-p^4", ")) \\ Charles R Greathouse IV, Jun 16 2011

a(n) = A000010(prime(n)^5). - R. J. Mathar, Oct 15 2017
From Amiram Eldar, Nov 22 2022: (Start)
a(n) = prime(n)^5 - prime(n)^4 = A050997(n) - A030514(n).
Product_{n>=1} (1 - 1/a(n)) = A065416. (End)

First Mathematica program corrected by Harvey P. Dale, Mar 29 2015

A146492 Decimal expansion of Product_{n>=2} (1 - 1/(n^4*(n-1))).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 13 2009

Product of Artin's constant of rank 4 and the equivalent almost-prime products.

			0.9298384739543468522383.. = (1-1/16)*(1-1/162)*(1-1/768)*(1-1/2500)*..

Cf. A065416.

r := 4 : 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[-5 + 11# - 6#^2 + #^3 & , k]]; RealDigits[Cosh[(Sqrt[3] Pi)/2]/(Pi g[1] g[2] g[3]) // Re, 10, 105] // First (* Jean-François Alcover, Feb 12 2013 *)
RealDigits[Re[N[Product[1 - 1/(n^4 (n - 1)), {n, 2, Infinity}], 110]]] (* Bruno Berselli, Apr 02 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 = 4.
s*Sum_{j=1..floor(s/5)} binomial(s-4j-1, j-1)/j = A058368(s)-1.
Equals 1/Product_{k=1..5} Gamma(1-x_k), where x_k are the 5 roots of the polynomial x*(x+1)^4-1. [R. J. Mathar, Feb 20 2009]

More terms from Jean-François Alcover, Feb 12 2013

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

A066834 Continued fraction expansion of Product_{p prime} (1 - 1/(p^5 - p^4)).

Original entry on oeis.org

0, 1, 13, 1, 1, 4, 1, 1, 1, 3, 16, 16, 1, 11, 1, 1, 1, 1, 9, 13, 1, 5, 22, 4, 2, 6, 1, 1, 1, 39, 1, 1, 3, 1, 12, 4, 1, 106, 15, 19, 7, 7, 4, 1, 5, 1, 2, 1, 1, 2, 4, 3, 1, 10, 1, 1, 1, 1, 2, 3, 1, 6, 7, 8, 1, 7, 5, 1, 2, 44, 2, 1, 5, 1, 4, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 20, 8, 2, 3, 1, 1, 1, 4, 1, 1, 2

Offset: 0

Randall L Rathbun, Jan 16 2002

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

Cf. A065416 (decimal expansion).

contfrac(prodeulerrat(1-1/(p^5-p^4))) \\ Amiram Eldar, Jun 13 2021

A146493 Decimal expansion of Product_{q in A001358} (1-1/(q^4*(q-1))).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 13 2009

Semiprime analog of A065416.

			0.998509500607573754... = (1-1/768)*(1-1/6480)*(1-1/52488)*..

The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_2(s)/j at r=4, where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.

A146494 Decimal expansion of Product_{q in A014612} (1-1/(q^4*(q-1))).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 13 2009

3-almost prime analog of A065416.

			0.99995964347639250716750... = (1-1/28672)*(1-1/228096)*(1-1/1784592)*...

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], table 3 at r=4, k=3.

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=4, where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.

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

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A065415 Decimal expansion of Product_{p prime} (1-1/(p^4-p^3)).

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A138407 a(n) = p^4*(p-1), where p = prime(n).

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A146492 Decimal expansion of Product_{n>=2} (1 - 1/(n^4*(n-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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A066834 Continued fraction expansion of Product_{p prime} (1 - 1/(p^5 - p^4)).

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A146493 Decimal expansion of Product_{q in A001358} (1-1/(q^4*(q-1))).

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A146494 Decimal expansion of Product_{q in A014612} (1-1/(q^4*(q-1))).

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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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