A001620 -id:A001620 - cp's OEIS frontend

A079381 Costé prime expansion of Euler's constant gamma (A001620).

2, 7, 13, 19, 89, 23, 11, 131, 73, 43, 37, 7, 11, 3, 3, 3, 3, 3, 5, 2, 7, 61, 251, 41, 13, 11, 7, 23, 29, 5, 13, 11, 3, 67, 29, 7, 5, 5, 2, 17, 5, 23, 7, 11, 2, 31, 29, 5, 5, 5, 3, 3, 5, 11, 5, 7, 7, 29, 17, 5, 2, 41, 13, 13, 11, 199, 157, 101, 37, 7, 127, 29, 11, 3, 3, 5, 17, 5, 7, 5, 2, 5

Offset: 0

N. J. A. Sloane, Feb 16 2003

For x in (0,1], define P(x) = min{p: p prime, 1/x < p}, Phi(x) = P(x)x - 1. Costé prime expansion of x(0) is sequence a(0), a(1), ... given by x(n) = Phi(x(n-1)) (n>0), a(n) = P(x(n)) (n >= 0).

G. C. Greubel, Table of n, a(n) for n = 0..2000
A. Costé, Sur un système fibré lié à la suite des nombres premiers, Exper. Math., 11 (2002), 383-405.

Cf. A079382, A079383, A079366-A079368.

Digits := 500: P := proc(x) local y; y := ceil(evalf(1/x)); if isprime(y) then y else nextprime(y); fi; end; F := proc(x) local y,i,t1; y := x; t1 := []; for i from 1 to 100 do p := P(y); t1 := [op(t1),p]; y := p*y-1; od; t1; end; F(gamma);

$MaxExtraPrecision = 500; P[x_] := Module[{y}, y = Ceiling[1/x]; If[PrimeQ[y], y, NextPrime[y]]]; F[x_] := Module[{y, i, t1}, y = x; t1 = {}; For[i = 1, i <= 100, i++, AppendTo[t1, p = P[y]]; y = p*y - 1]; t1]; F[EulerGamma] (* G. C. Greubel, Jan 20 2019 *)

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 17 2003

A097668 Decimal expansion of the constant 5*exp(psi(2/5)+EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-5 linear recursions with varying coefficients (see A097680 for example).

			c = 0.68747420699080196070816422133398475499777353078320593237327...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 0..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A097663-A097667, A097669-A097676.

RealDigits[ GoldenRatio^(Sqrt[5]/2)/5^(1/4)*E^(-Pi/2Sqrt[1 - 2/Sqrt[5]]), 10, 105][[1]] (* Robert G. Wilson v, Aug 27 2004 *)

PARI
```
5*exp(psi(2/5)+Euler)
```

c = ((sqrt(5)+1)/2)^(sqrt(5)/2)/5^(1/4)*exp(-Pi/2*sqrt(1-2/sqrt(5))).

More terms from Robert G. Wilson v, Aug 27 2004

A097669 Decimal expansion of the constant 5*exp(psi(3/5)+EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-5 linear recursions with varying coefficients (see A097680 for example).

			c = 1.90795953254354252255333813972952036908516068359082968228223...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A097663-A097668, A097670-A097676.

RealDigits[ GoldenRatio^(Sqrt[5]/2)/5^(1/4)*E^(Pi/2Sqrt[1 - 2/Sqrt[5]]), 10, 105][[1]] (* Robert G. Wilson v, Aug 27 2004 *)

PARI
```
5*exp(psi(3/5)+Euler)
```

c = ((sqrt(5)+1)/2)^(sqrt(5)/2)/5^(1/4)*exp(Pi/2*sqrt(1-2/sqrt(5))).

More terms from Robert G. Wilson v, Aug 27 2004

A097670 Decimal expansion of the constant 5*exp(psi(4/5) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-5 linear recursions with varying coefficients (see A097680 for example).

			c = 3.39276427892784980770475505655447128392740109258608442234780...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A097663-A097669, A097671-A097676.

RealDigits[ GoldenRatio^(-Sqrt[5]/2)/5^(1/4)*E^(Pi/2Sqrt[1 + 2/Sqrt[5]]), 10, 105][[1]] (* Robert G. Wilson v, Aug 27 2004 *)

PARI
```
5*exp(psi(4/5)+Euler)
```

c = ((sqrt(5)+1)/2)^(-sqrt(5)/2)/5^(1/4)*exp(Pi/2*sqrt(1+2/sqrt(5))).

More terms from Robert G. Wilson v, Aug 27 2004

A097672 Decimal expansion of the constant 6*exp(psi(5/6) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-6 linear recursions with varying coefficients (see A097681 for example).

			c = 4.38524598625080817369948994322956207765080128500902763042006...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A097663-A097671, A097673-A097676.

RealDigits[1/Sqrt[12]*E^(Pi/2Sqrt[3]), 10, 105][[1]] (* Robert G. Wilson v, Aug 27 2004 *)

PARI
```
6*exp(psi(5/6)+Euler)
```

c = 1/sqrt(12)*exp(Pi/2*sqrt(3)).

More terms from Robert G. Wilson v, Aug 27 2004

A097674 Decimal expansion of the constant 8*exp(psi(3/8) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-8 linear recursions with varying coefficients (see A097682 for example).

			c = 0.90724581788216460753879452479208121378777525423587495906871...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 0..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A097663-A097673, A097675-A097676.

RealDigits[(1 + Sqrt[2])^(Sqrt[2])/2E^(-Pi/2*(Sqrt[2] - 1)), 10, 105][[1]] (* Robert G. Wilson v, Aug 27 2004 *)

PARI
```
8*exp(psi(3/8)+Euler)
```

c = (1+sqrt(2))^(sqrt(2))/2*exp(-Pi/2*(sqrt(2)-1)).

More terms from Robert G. Wilson v, Aug 27 2004

A097675 Decimal expansion of the constant 8*exp(psi(5/8) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-8 linear recursions with varying coefficients (see A097682 for example).

			c = 3.33325212658541721540039076972102211743980259727655469662827...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A097663-A097674, A097676.

RealDigits[(1 + Sqrt[2])^(Sqrt[2])/2E^(Pi/2*(Sqrt[2] - 1)), 10, 105][[1]] (* Robert G. Wilson v, Aug 27 2004 *)

PARI
```
8*exp(psi(5/8)+Euler)
```

c = (1+sqrt(2))^(sqrt(2))/2*exp(Pi/2*(sqrt(2)-1)).

More terms from Robert G. Wilson v, Aug 27 2004

A279161 Define P = e^gamma*loglog(n) and Q = 3/loglog(n), where gamma is Euler's constant A001620. Then a(n) = phi(n) - ceiling(n/(P + Q)), where phi(n) is Euler's function A000010.

Original entry on oeis.org

1, 1, 3, 1, 4, 2, 4, 2, 7, 1, 9, 3, 4, 4, 12, 2, 13, 3, 7, 5, 17, 2, 14, 6, 12, 5, 21, 1, 23, 9, 12, 8, 16, 4, 27, 9, 15, 7, 31, 2, 32, 10, 14, 12, 35, 5, 31, 9, 20, 12, 40, 6, 28, 11, 23, 15, 45, 3, 46, 16, 22, 18, 34, 5, 51, 17, 29, 8, 54, 8, 56, 20, 23, 19

Offset: 3

Vladimir Shevelev, Dec 07 2016

nonn

The best known lower estimate for phi(n)is phi(n) > n/(P + Q), n >= 3 [Rosser and Schoenfeld] (and, for each eps > 0, there exist infinitely many n for which phi(n) < n/P', where in P' e^gamma is replaced by e^(gamma-eps) [Landau]). So a(n) >= 0.

E. Landau, Handbuch der Lehre yon der Verteilung der Primzahlen, 2 vols., Leipzig, Teubner, 1909. Reprinted in 1953 by Chelsea Publishing Co., New York.

Peter J. C. Moses, Table of n, a(n) for n = 3..5002
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), pp. 64-94.

Cf. A000010, A001620.

a(n)=my(LL=log(log(n)),P=LL*exp(Euler),Q=3/LL); eulerphi(n) - ceil(n/(P+Q)) \\ Charles R Greathouse IV, Dec 07 2016

More terms from Peter J. C. Moses, Dec 07 2016

A306765 Decimal expansion of lim_{k->oo} (k^A001620 / k!) * Product_{j=1..k} Gamma(1/j).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Mar 08 2019

			2.0344489454876164779803555318869026355979439863702376260005284165650078277571...

Vaclav Kotesovec, Table of n, a(n) for n = 1..297

Cf. A001620, A231132, A303670, A306760, A306769.

evalf(exp(-gamma^2 + Sum((-1)^j*Zeta(j)^2/j, j=2..infinity)), 100);

slogam = Table[Sum[LogGamma[1/j], {j, 1, n}], {n, 1, 1000}]; $MaxExtraPrecision = 1000; funs[n_] := E^slogam[[n]] * n^EulerGamma/n!; Do[Print[N[Sum[(-1)^(m + j) * funs[j*Floor[Length[slogam]/m]] * (j^(m - 1)/(j - 1)!/(m - j)!), {j, 1, m}], 80]], {m, 10, 100, 10}]

exp(-Euler^2 + sumalt(j=2, (-1)^j*zeta(j)^2/j))

Equals exp(-gamma^2 + Sum_{j>=2} (-1)^j*Zeta(j)^2/j), where gamma is the Euler-Mascheroni constant A001620.
Equals exp(-gamma^2 + A306769).
Equals lim_{k->oo} k^(k*(2*k+1) + 2*gamma) * (2*Pi)^k * exp(1/6 + log(k)^2 - 2*k^2) / A306760(k).

A059559 Beatty sequence for 1 + log(1/gamma), (gamma is the Euler-Mascheroni constant A001620).

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 72, 74, 75, 77, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 96, 97, 99, 100, 102, 103, 105, 106

Offset: 1

Mitch Harris, Jan 22 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt, Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Beatty complement is A059560.

R:=RealField(100); [Floor(1+Log(1/EulerGamma(R))*n): n in [1..100]]; // G. C. Greubel, Aug 27 2018

Table[Floor[n*(1 + Log[1/EulerGamma])], {n,1,100}] (* G. C. Greubel, Aug 27 2018 *)

{ default(realprecision, 100); b=1 + log(1/Euler); for (n = 1, 2000, write("b059559.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = floor(n*(1 + log(1/Euler))). - Michel Marcus, Jan 05 2015

Corrected the definition from 1-log(1/gamma) to 1+log(1/gamma). - Harry J. Smith, Jun 28 2009

cp's OEIS Frontend

A079381 Costé prime expansion of Euler's constant gamma (A001620).

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A097668 Decimal expansion of the constant 5*exp(psi(2/5)+EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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A097669 Decimal expansion of the constant 5*exp(psi(3/5)+EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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A097670 Decimal expansion of the constant 5*exp(psi(4/5) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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A097672 Decimal expansion of the constant 6*exp(psi(5/6) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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A097674 Decimal expansion of the constant 8*exp(psi(3/8) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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