A200064 -id:A200064 - cp's OEIS frontend

A256843 Decimal expansion of the generalized Euler constant gamma(2,3).

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

Offset: 0

Jean-François Alcover, Apr 11 2015

			0.07320737570615959366903185990752913907462383026830934562939...

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. H. Lehmer, Euler constants for arithmetic progressions, Acta Arith. 27 (1975), p. 134.

Cf. A001620 (gamma(1,1) = EulerGamma), A002391, A200064.
Primitive ruler-and-compass constructible gamma(r,k): A228725 (1,2), A256425 (1,3), A256778 (1,4), A256779 (1,5), A256780 (2,5), A256781 (1,8), A256782 (3,8), A256783 (1,12), A256784 (5,12).
Other gamma(r,k) (1 <= r <= k <= 5): A239097 (2,2),  A256843 (2,3),  A256844 (3,3), A256845 (2,4), A256846 (3,4), A256847 (4,4), A256848 (3,5), A256849 (4,5), A256850 (5,5).

SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/3 - Pi(R)/(6*Sqrt(3)) + Log(3)/6; // G. C. Greubel, Aug 28 2018

Join[{0}, RealDigits[-Log[3]/3 - PolyGamma[2/3]/3, 10, 105] // First]

default(realprecision, 100); Euler/3 - Pi/(6*sqrt(3)) + log(3)/6 \\ G. C. Greubel, Aug 28 2018

Equals EulerGamma/3 - Pi/(6*sqrt(3)) + log(3)/6.

Equals -(psi(2/3) + log(3))/3 = (A200064 - A002391)/3. - Amiram Eldar, Jan 07 2024

A200135 Decimal expansion of the negated value of the digamma function at 1/5.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			Psi(1/5) =  -5.289039896592188295547207962...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Digamma function
Index entries for sequences related to the digamma function

Cf. A020759, A047787, A020777, A200064, A200134, A200136, A200137, A200138.

SetDefaultRealField(RealField(100)); R:= RealField(); -EulerGamma(R) -Pi(R)*Sqrt(1+2/Sqrt(5))/2 -5*Log(5)/4 -Sqrt(5)/4*Log((3+Sqrt(5)/2) ); // G. C. Greubel, Sep 03 2018

-gamma-Pi*sqrt(1+2/sqrt(5))/2-5*log(5)/4-sqrt(5)/4*log((3+sqrt(5)/2) ); evalf(%) ;

RealDigits[-PolyGamma[1/5], 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

-psi(1/5) \\ Charles R Greathouse IV, Jul 19 2013

Psi(1/5) = -gamma - Pi*sqrt(1 + 2/sqrt(5))/2 - 5*log(5)/4 -sqrt(5)*log((3 + sqrt(5))/2)/4 where gamma = A001620, sqrt(1 + 2/sqrt(5)) = A019952, (3 + sqrt(5))/2 = A104457.

More terms from Jean-François Alcover, Feb 11 2013

A222457 Decimal expansion of the negated value of the digamma function at 1/6.

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Feb 21 2013

			Psi(1/6) = -6.3321275053749147924249615748457777225904948...

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Ernst D. Krupnikov and K. S. Kolbig, Some special cases of the generalized hypergeometric function _{q+1}F_q, J. Comp. Appl. Math. 78 (1997) 79-95
Eric Weisstein's World of Mathematics, Gauss's Digamma Theorem.
Wikipedia, Digamma function: special values.
Index entries for sequences related to the digamma function

Cf. A020759, A047787, A200064, A222458.

RealDigits[-PolyGamma[1/6], 10, 90][[1]]

Maxima
```
fpprec:90; ev(bfloat(-psi[0](1/6)));
```
PARI
```
-psi(1/6)
```

Psi(1/6) = -gamma -Pi*sqrt(3)/2 -3*log(3)/2 -2*log(2).

A222458 Decimal expansion of the negated value of the digamma function at 5/6.

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Feb 21 2013

			Psi(5/6) = -0.890729412672261240642726801919310525738296...

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Gauss's Digamma Theorem.
Wikipedia, Digamma function: Gauss's digamma theorem.
Index entries for sequences related to the digamma function

Cf. A020759, A047787, A200064, A222457.

RealDigits[-PolyGamma[5/6], 10, 90][[1]]

Maxima
```
fpprec:90; ev(bfloat(-psi[0](5/6)));
```
PARI
```
-psi(5/6)
```

Psi(5/6) = -gamma + Pi*sqrt(3)/2 - 3*log(3)/2 - 2*log(2).

A250129 Decimal expansion of the negated value of the digamma function at 1/8.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jan 15 2015

			Psi(1/8) = -8.388492663295854867802742923086343000051446...

Eric Weisstein's MathWorld, Gauss's Digamma Theorem.
Wikipedia, Digamma function: special values.
Index entries for sequences related to the digamma function

Cf. A020759, A020777, A047787, A200064, A222457, A222458.

RealDigits[PolyGamma[1/8], 10, 105] // First

-psi(1/8) \\ Charles R Greathouse IV, Jan 15 2015

Psi(1/8) = -gamma - (1/2)*(1+sqrt(2))*Pi - sqrt(2)*arccoth(sqrt(2)) - 4*log(2).

A200136 Decimal expansion of the negated value of the digamma function at 2/5.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			Psi(2/5) = -2.5613845445851161457306754820475...

Index entries for sequences related to the digamma function

Cf. A020759, A020777, A047787, A200064, A200135-A200138.

-gamma-Pi*sqrt(1-2/sqrt(5))/2-5*log(5)/4+sqrt(5)*log((3+sqrt(5))/2)/4 ; evalf(%) ;

RealDigits[ PolyGamma[2/5], 10, 84] // First (* Jean-François Alcover, Feb 21 2013 *)

-psi(2/5) \\ Charles R Greathouse IV, Jul 19 2013

Psi(2/5) = -gamma -Pi*sqrt(1-2/sqrt 5)/2 -5*log(5)/4 +sqrt(5)*log((3+sqrt 5)/2)/4.

A200063 Indices n where A079878(n) = n.

Original entry on oeis.org

1, 2, 8, 32, 46, 392, 12230, 155942, 659488, 1025582, 1047128, 3437088, 1449322158, 1452777560, 1691887144, 4558298126, 4840156480, 39554086678, 353617531486, 608231808384, 619986226720, 969355365422

Offset: 1

R. J. Mathar, Nov 13 2011

nonn

a(23) > 4*10^12. - Donovan Johnson, Nov 21 2011

			A079878(46)=46, which adds 46 to the sequence.

Cf. A200064

import Data.List (elemIndices)
a200063 n = a200063_list !! (n-1)
a200063_list = map (+ 1) $ elemIndices 0 $ zipWith (-) [1..] a079878_list
-- Reinhard Zumkeller, Nov 13 2011

{n: A079878(n)=n}.

a(13)-a(22) from Donovan Johnson, Nov 21 2011

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A256843 Decimal expansion of the generalized Euler constant gamma(2,3).

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A200135 Decimal expansion of the negated value of the digamma function at 1/5.

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A222457 Decimal expansion of the negated value of the digamma function at 1/6.

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A222458 Decimal expansion of the negated value of the digamma function at 5/6.

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A250129 Decimal expansion of the negated value of the digamma function at 1/8.

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A200136 Decimal expansion of the negated value of the digamma function at 2/5.

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A200063 Indices n where A079878(n) = n.

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