A001620 -id:A001620 - cp's OEIS frontend

A059557 Beatty sequence for 1 + gamma^2, (gamma is the Euler-Mascheroni constant A001620).

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94

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 A059558.

R:=RealField(100); [Floor((1+EulerGamma(R)^2)*n): n in [1..100]]; // G. C. Greubel, Aug 27 2018

Table[Floor[(1 + EulerGamma^2)*n], {n,1,100}] (* G. C. Greubel, Aug 27 2018 *)

{ default(realprecision, 100); b=1 + Euler^2; for (n = 1, 2000, write("b059557.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = A042968(n-1), 1<=n<2146. - R. J. Mathar, Oct 05 2008

A059565 Beatty sequence for e^gamma (gamma is the Euler-Mascheroni constant A001620).

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 14, 16, 17, 19, 21, 23, 24, 26, 28, 30, 32, 33, 35, 37, 39, 40, 42, 44, 46, 48, 49, 51, 53, 55, 56, 58, 60, 62, 64, 65, 67, 69, 71, 73, 74, 76, 78, 80, 81, 83, 85, 87, 89, 90, 92, 94, 96, 97, 99, 101, 103, 105, 106, 108, 110, 112, 113, 115, 117

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

Cf. A073004. Beatty complement is A059566.

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

Table[ Floor[ n * E^EulerGamma], {n, 1, 70} ]

{ default(realprecision, 100); b=exp(1)^Euler; for (n = 1, 2000, write("b059565.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

A097664 Decimal expansion of the constant 3*exp(psi(2/3) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

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-3 linear recursions with varying coefficients (see A097678 for example).

			c = 1.42988430840123420566179042477513809656498236767564464887634...

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.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.

Cf. A097663, A097665-A097676.

SetDefaultRealField(RealField(100)); R:= RealField(); (1/Sqrt(3))*Exp(Pi(R)/Sqrt(12)); // G. C. Greubel, Aug 27 2018

RealDigits[1/Sqrt[3]*E^(Pi/Sqrt[12]), 10, 105][[1]] (* Robert G. Wilson v, Aug 28 2004 *)

PARI
```
3*exp(psi(2/3)+Euler)
```

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

More terms from Robert G. Wilson v, Aug 28 2004

A097666 Decimal expansion of the constant 4*exp(psi(3/4) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

2, 4, 0, 5, 2, 3, 8, 6, 9, 0, 4, 8, 2, 6, 7, 5, 8, 2, 7, 7, 3, 6, 5, 1, 7, 8, 3, 3, 3, 5, 1, 9, 1, 6, 5, 6, 3, 1, 9, 5, 0, 8, 5, 4, 3, 7, 3, 3, 2, 2, 6, 7, 4, 7, 0, 0, 1, 0, 4, 0, 7, 7, 4, 4, 6, 2, 1, 2, 7, 5, 9, 5, 2, 4, 4, 5, 7, 9, 1, 0, 6, 8, 3, 7, 4, 3, 5, 2, 3, 8, 3, 2, 9, 1, 9, 4, 1, 6, 7, 7, 3, 2, 8, 6, 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-4 linear recursions with varying coefficients (see A097679 for example).

			c = 2.40523869048267582773651783335191656319508543733226747001040...

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-A097665, A097667-A097676.

SetDefaultRealField(RealField(100)); R:= RealField(); Exp(Pi(R)/2)/2; // G. C. Greubel, Sep 07 2018

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

PARI
```
4*exp(psi(3/4)+Euler)
```

c = 1/2*exp(Pi/2).

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

A097667 Decimal expansion of the constant 5*exp(psi(1/5) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

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 = 0.04494182877920882060846739642766520340238594371059869805861...

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-A097666, A097668-A097676.

RealDigits[ GoldenRatio^(-Sqrt[5]/2)/5^(1/4)*E^(-Pi/2*Sqrt[1 + 2/Sqrt[5]]), 10, 104][[1]] (* Robert G. Wilson v, Aug 27 2004 *)
Join[{0}, RealDigits[N[5*Exp[PolyGamma[1/5] + EulerGamma], 120], 10, 100][[1]]] (* G. C. Greubel, Dec 31 2016 *)

PARI
```
5*exp(psi(1/5)+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

A097671 Decimal expansion of the constant 6*exp(psi(1/6) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

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-6 linear recursions with varying coefficients (see A097681 for example).

			c = 0.01900311489814044762029209432973427009446270150034137604224...

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-A097670, A097672-A097676.

RealDigits[1/Sqrt[12]*E^(-Pi/2Sqrt[3]), 10, 104][[1]] (* Robert G. Wilson v, Aug 27 2004 *)

PARI
```
6*exp(psi(1/6)+Euler)
```

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

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

A097673 Decimal expansion of the constant 8*exp(psi(1/8) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

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 = 0.00324112283009630737475117121791901701073847922151040069299...

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-A097672, A097674-A097676.

RealDigits[(1 + Sqrt[2])^(-Sqrt[2])/2E^(-Pi/2*(1 + Sqrt[2])), 10, 103][[1]] (* Robert G. Wilson v, Aug 27 2004 *)

PARI
```
8*exp(psi(1/8)+Euler)
```

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

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

A098967 Write down decimal expansion of Euler-Mascheroni constant gamma (A001620); divide up into chunks of minimal length so that chunks are increasing numbers and do not begin with 0.

Original entry on oeis.org

5, 7, 72, 156, 649, 1532, 8606, 65120, 90082, 402431, 421593, 3593992, 3598805, 7672348, 8486772, 67776646, 70936947, 632917467, 4951463144, 7249807082, 48096050401, 448654283622, 4173997644923, 5362535003337, 42937337737673

Offset: 0

Sam Handler (shandler(AT)Macalester.edu), Oct 25 2004

			0.57721566490153286060651209008240243104215933593992359880576723488...

G. C. Greubel, Table of n, a(n) for n = 0..1500

Cf. A001620, A002852, A059856, A033091.

f[n_] := Block[{ts = StringDrop[ ToString[ N[n, 250]], 2], a = {}, d = 0, k = 1}, While[ ToExpression[ts] > d, While[d >= ToExpression[ StringTake[ts, k]], k++ ]; te = ToExpression[ StringTake[ts, k]]; d = te; AppendTo[a, te]; ts = StringDrop[ts, k]; If[k > 1, k-- ]]; a]; f[EulerGamma] (* Robert G. Wilson v, Nov 01 2004 *)

Corrected and extended by Robert G. Wilson v, Nov 01 2004

A345208 Decimal expansion of log(2*Pi) - gamma - 1, where gamma is Euler's constant (A001620).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 10 2021

The first two formulae (in the Formula section) are similar to the sum and integral lim_{n->oo} (1/n) * Sum_{k=1..n} frac(n/k) = Integral_{x=0..1} frac(1/x) dx = 1 - gamma (A153810).

The second raw moment of the distribution of the fractional part of 1/x, where x is chosen uniformly at random from (0, 1]. Since the expected value is 1 - gamma, the second central moment, or variance, is log(2*Pi) - gamma - 1 - (1 - gamma)^2 = log(2*Pi) - gamma^2 + gamma - 2 = 0.081914807503... and the standard deviation is sqrt(log(2*Pi) - gamma^2 + gamma - 2) = 0.2862076300...

			0.26066140150781262295414738272883284868063561133564...

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See Problem 3.42, pages 145 and 195.

Ovidiu Furdui, Solution to Problem U27, Mathematical Reflections, Vol. 6 (2006), pp. 27-28.
Ovidiu Furdui, Exotic fractional part integrals and Euler's constant, Analysis, Vol. 31, No. 3 (2011), pp. 249-257.
Mircea Ivan and Alexandru Lupaş, Problem 11206, The American Mathematical Monthly, Vol. 113, No. 2 (2006), p. 180; A Limit Involving Euler's Constant, Solution to problem 11206 by Richard A. Stong, ibid., Vol. 114, No. 10 (2007), pp. 928-929.
Albert F. S. Wong, Problem 1845, Mathematics Magazine, Vol. 83, No. 2 (2010), p. 150; Integrating a square-fractional-reciprocal function, Solution to problem 1845 by Allen Stenger, ibid., Vol. 84, No. 2 (2011), pp. 155-156.

Cf. A001008, A001620, A002805, A061444, A153810.

RealDigits[Log[2*Pi] - EulerGamma - 1, 10, 100][[1]]

Equals lim_{n->oo} (1/n) * Sum_{k=1..n} frac(n/k)^2, where frac(x) = x - floor(x) is the fractional part of x.
Equals Integral_{x=0..1} frac(1/x)^2 dx.
Equals 2 * Sum_{k>=2} (zeta(k)-1)/(k*(k+1)).
Equals A061444 - A001620 - 1.
Equals -2 * Sum_{k>=1} (H(k) - log(k) - gamma - 1/(2*k)), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Furdui, 2013). - Amiram Eldar, Mar 26 2022

A053977 Engel expansion of the Euler-Mascheroni constant gamma A001620 = 0.57721566... .

Original entry on oeis.org

2, 7, 13, 19, 85, 2601, 9602, 46268, 4812284, 147961485, 210810243, 814960948, 1003849128, 1016803038, 12917183059, 26242325020, 22215291139324, 30797877759859, 60139200644343, 121848657453426, 133555928335475

Offset: 1

Jeppe Stig Nielsen, Apr 02 2000

Cf. A006784 for definition of Engel expansion.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

T. D. Noe, Table of n, a(n) for n=1..300
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Eric Weisstein's World of Mathematics, Engel Expansion
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant
Index entries for sequences related to Engel expansions

Cf. A006784, A028254, A028257.

EngelExp[ A_, n_ ] := Join[ Array[ 1&, Floor[ A ] ], First@Transpose@NestList[ {Ceiling[ 1/Expand[ #[ [ 1 ] ]#[ [ 2 ] ]-1 ] ], Expand[ #[ [ 1 ] ]#[ [ 2 ] ]-1 ]}&, {Ceiling[ 1/(A-Floor[ A ]) ], A-Floor[ A ]}, n-1 ] ]

More terms and additional comments from Mitch Harris, Jan 15 2001

cp's OEIS Frontend

A059557 Beatty sequence for 1 + gamma^2, (gamma is the Euler-Mascheroni constant A001620).

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A059565 Beatty sequence for e^gamma (gamma is the Euler-Mascheroni constant A001620).

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A097664 Decimal expansion of the constant 3*exp(psi(2/3) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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A097666 Decimal expansion of the constant 4*exp(psi(3/4) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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Magma

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A097667 Decimal expansion of the constant 5*exp(psi(1/5) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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Programs

Mathematica

PARI

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A097671 Decimal expansion of the constant 6*exp(psi(1/6) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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Programs

Mathematica

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