A002210 -id:A002210 - cp's OEIS frontend

A064994 A Beatty sequence from Khinchin's constant (A002210).

1, 3, 5, 6, 8, 10, 11, 13, 15, 16, 18, 20, 21, 23, 25, 26, 28, 30, 32, 33, 35, 37, 38, 40, 42, 43, 45, 47, 48, 50, 52, 53, 55, 57, 58, 60, 62, 64, 65, 67, 69, 70, 72, 74, 75, 77, 79, 80, 82, 84, 85, 87, 89, 91, 92, 94, 96, 97, 99, 101, 102, 104, 106, 107, 109, 111, 112, 114

Offset: 1

Robert G. Wilson v, Oct 31 2001

nonn

Cf. A002210, A064995.

k = N[Khinchin, 100]; Table[ Floor[ n*(k - 1) ], {n, 1, 70} ]

A064995 A Beatty sequence from Khintchin's constant (A002210).

Original entry on oeis.org

2, 4, 7, 9, 12, 14, 17, 19, 22, 24, 27, 29, 31, 34, 36, 39, 41, 44, 46, 49, 51, 54, 56, 59, 61, 63, 66, 68, 71, 73, 76, 78, 81, 83, 86, 88, 90, 93, 95, 98, 100, 103, 105, 108, 110, 113, 115, 118, 120, 122, 125, 127, 130, 132, 135, 137, 140, 142, 145, 147, 149, 152

Offset: 1

Robert G. Wilson v, Oct 31 2001

nonn

Cf. A002210, A064994.

k = N[Khinchin, 100]; Table[ Floor[ n*(k - 1)/(k - 2) ], {n, 1, 70} ]

A100485 Decimal expansion of: (1) a simple-continued-fraction-like nesting in which all "partial quotients" are Khinchin's constant (A002210), or, equivalently, (2) the positive solution p of the polynomial p^2 - Khinchin*p - 1 = 0.

Original entry on oeis.org

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

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 22 2004

			3.016916286165207794242 437240471558379172667 864562044748908962043 759189468942531844688 22650236967025258672...

Cf. A002210.

N[FromContinuedFraction[{{Khinchin}}], 105]

(Khinchin + (4+Khinchin*Khinchin)^(1/2))/2

A119928 Continued fraction expansion of the value of Minkowski's question mark function at Khinchin's constant (A002210).

Original entry on oeis.org

2, 1, 3, 11, 10, 2, 1, 1, 2, 5, 2, 4, 23, 45, 1, 3, 9, 4, 3, 1, 2, 1, 24, 13, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 2, 51, 2, 7, 2, 1198400, 1, 3, 2, 10, 1, 11, 13, 1, 2, 1, 4, 1, 10, 3, 13, 1, 1, 1, 2, 13, 1, 1, 122, 148, 1, 48, 3, 1, 46, 1, 1, 1, 4, 2, 1, 5, 4, 1, 2, 1, 1, 8, 1, 8, 5, 1, 7, 1, 2, 1, 1

Offset: 0

Joseph Biberstine (jrbibers(AT)indiana.edu), May 29 2006; corrected Jun 04 2006

Cf. A119929.

ContinuedFraction[(cf = ContinuedFraction[Khinchin, 80(*arbitrary precision*)]; IntegerPart[Khinchin] + Sum[(-1)^(k)/2^(Sum[cf[[i]], {i, 2, k}] - 1), {k, 2, Length[cf]}])]

A119929 Decimal expansion of the value of Minkowski's question mark function at Khinchin's constant (A002210).

Original entry on oeis.org

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

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), May 29 2006; corrected Jun 04 2006

			2.755508409987669440025291969515591761208384014026394889775...

Index entries for sequences related to Minkowski's question mark function

Cf. A119928.

(*ensure variables are appropriately Cleared*) Off[ContinuedFraction::incomp]; mq[x_] := (If[Element[x, Rationals], cf = ContinuedFraction[x], cf = ContinuedFraction[x, 80(*arbitrary precision*)]]; IntegerPart[x] + Sum[(-1)^(k)/2^(Sum[cf[[i]], {i, 2, k}] - 1), {k, 2, Length[cf]}]); RealDigits[mq[Khinchin],10]
RealDigits[(cf = ContinuedFraction[Khinchin, 80(*arbitrary precision*)]; IntegerPart[Khinchin] + Sum[(-1)^(k)/2^(Sum[cf[[i]], {i, 2, k}] - 1), {k,2, Length[cf]}]), 10]

A100611 Continued fraction expansion of (Khinchin + (4+Khinchin*Khinchin)^(1/2))/2 [A100485], where Khinchin is Khinchin's constant (A002210).

Original entry on oeis.org

3, 59, 8, 1, 2, 1, 1, 1, 1, 1, 2, 6, 1, 1, 3, 2, 1, 7, 1, 3, 2, 8, 5, 1, 3, 2, 6, 1, 2, 2, 2, 7, 1, 1, 3, 1, 10, 3, 1, 1, 3, 1, 1, 1, 2, 2, 3, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 5, 9, 2, 1, 21, 3, 1, 1, 2, 1, 13, 1, 3, 1, 1, 1, 1, 2, 2, 1, 3, 5, 1, 1, 2, 1, 3, 1, 1, 6, 4, 2, 1, 2, 1, 2, 9, 5, 16, 4, 1, 2, 2, 4

Offset: 0

Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 01 2004

The geometric mean of the first 5000 partial quotients (Product[a(n),{n,1,5000}]^(1/5000)), approximately 2.6289, suggests this constant's partial quotients do indeed converge to Khinchin's constant.

Cf. A100485 (decimal expansion), A002210.

ContinuedFraction[(Khinchin + (4+Khinchin*Khinchin)^(1/2))/2, 130]

Offset changed by Andrew Howroyd, Aug 04 2024

A175819 Partial sums of digits of decimal expansion of Khinchin's constant (sequence A002210).

Original entry on oeis.org

2, 8, 16, 21, 25, 30, 32, 32, 32, 33, 33, 39, 44, 47, 47, 53, 57, 61, 66, 69, 69, 78, 85, 86, 90, 98, 101, 106, 110, 118, 119, 126, 135, 140, 146, 155, 158, 166, 168, 168, 171, 179, 181, 183, 192, 195, 204, 213, 217, 221, 227, 229, 238, 243, 246, 246, 251, 252, 253

Offset: 1

Michel Lagneau, Sep 11 2010

			Khinchin's constant 2.6854520010 ... so the sums are 2, 2+6, 2+6+8, 2+6+8+5, 2+6+8+5+4..., leading to the terms 2, 8, 16, 21, 25,...

Cf. A002210 for digits of Khintchine's constant

L= Rest@FoldList[ Plus, 0, First@ RealDigits[Khinchin, 10, 100]]
Accumulate[RealDigits[Khinchin,10,60][[1]]]  (* Harvey P. Dale, Mar 24 2011 *)

A131688 Decimal expansion of the constant Sum_{k>=1} log(k + 1) / (k * (k + 1)).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Sep 14 2007

Given A131385(n) = Product_{k=1..n} floor((n+k)/k), then limit A131385(n+1)/A131385(n) = exp(c), where c = this constant. - Paul D. Hanna, Nov 26 2012

Closely related to A085361 (the exponent in the definition of A085291). - Yuriy Sibirmovsky, Sep 04 2016

			1.257746886944369630009899830495881528511540890508884868977540833522...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, page 62. [Jean-François Alcover, Mar 21 2013]

G. C. Greubel, Table of n, a(n) for n = 1..1000
Khristo N. Boyadzhiev, A special constant and series with zeta values and harmonic numbers, arXiv:1903.11141 [math.NT], 2019.
Mark W. Coffey, Series of zeta values, the Stieltjes constants and a sum S_gamma(n), arXiv:math-ph/0706.0345, 2007-2009, eq (38a).
Paul Erdős, S. W. Graham, Aleksandar Ivic and Carl Pomerance, On the number of divisors of n!, Analytic Number Theory, Volume 138, Progress in Mathematics pp 337-355.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 538.
Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Volume 29, Issue 2, June 1988, Pages 196-205.

Cf. A000005, A002210, A027423, A075887, A131385, A244109, A001620, A085361, A345682.

SetDefaultRealField(RealField(100)); L:=RiemannZeta(); (&+[(-1)^(n+1)*Evaluate(L,n+1)/n: n in [1..10^3]]); // G. C. Greubel, Nov 15 2018

evalf(sum((-1)^(n+1)*Zeta(n+1)/n, n=1..infinity), 120); # Vaclav Kotesovec, Dec 11 2015
evalf(Sum(-Zeta(1, k), k = 2..infinity), 120); # Vaclav Kotesovec, Jun 18 2021

Sum[ -Zeta'[1 + k], {k, 1, Infinity}] (* Vladimir Reshetnikov, Dec 28 2008 *)
Integrate[EulerGamma/x + PolyGamma[0, 1+x]/x, {x, 0, 1}] // N[#, 105]& // RealDigits[#][[1]]& (* or *) Integrate[x*Log[x]/((1-x)*Log[1-x]), {x, 0, 1}] // N[#, 105]& // RealDigits[#][[1]]& (* Jean-François Alcover, Feb 04 2013 *)
$MaxExtraPrecision = 200; NIntegrate[HarmonicNumber[t]/t, {t, 0, 1}, WorkingPrecision -> 105] (* Yuriy Sibirmovsky, Sep 04 2016 *)
digits = 120; RealDigits[NSum[(-1)^(n + 1)*Zeta[n + 1]/n, {n,1,Infinity}, NSumTerms -> 20*digits, WorkingPrecision -> 10*digits, Method -> "AlternatingSigns"], 10, digits][[1]] (* G. C. Greubel, Nov 15 2018 *)

PARI
```
sumalt(s=1, (-1)^(s+1)/s*zeta(s+1) )
```

suminf(k=2, -zeta'(k)) \\ Vaclav Kotesovec, Jun 17 2021

numerical_approx(sum((-1)^(k+1)*zeta(k+1)/k for k in [1..1000]), digits=100) # G. C. Greubel, Nov 15 2018

Equals Sum_{s>=1} (-1)^(s+1)*zeta(s+1)/s.
Equals Sum_{k>=1} -zeta'(1 + k), where Zeta' is the derivative of the Riemann zeta function. - Vladimir Reshetnikov, Dec 28 2008
Equals Sum_{s>=1} log(1+1/s)/s. - Jean-François Alcover, Mar 26 2013
Equals Integral_{t=0..1} H(t)/t dt. Compare to A001620 = Integral_{t=0..1} H(t) dt. Where H(t) are generalized harmonic numbers. - Yuriy Sibirmovsky, Sep 04 2016
Equals lim_{n->oo} log(d(n!))*log(n)/n, where d(n) is the number of divisors of n (A000005) (Erdős et al., 1996). - Amiram Eldar, Nov 07 2020

Extended to 105 digits by Jean-François Alcover, Feb 04 2013

A002211 Continued fraction for Khintchine's constant.

Original entry on oeis.org

2, 1, 2, 5, 1, 1, 2, 1, 1, 3, 10, 2, 1, 3, 2, 24, 1, 3, 2, 3, 1, 1, 1, 90, 2, 1, 12, 1, 1, 1, 1, 5, 2, 6, 1, 6, 3, 1, 1, 2, 5, 2, 1, 2, 1, 1, 4, 1, 2, 2, 3, 2, 1, 1, 4, 1, 1, 2, 5, 2, 1, 1, 3, 29, 8, 3, 1, 4, 3, 1, 10, 50, 1, 2, 2, 7, 6, 2, 2, 16, 4, 4, 2, 2, 3, 1, 1, 7, 1, 5, 1, 2, 1, 5, 3, 1

Offset: 0

N. J. A. Sloane

Incrementally larger terms in the continued fraction for Khintchine's constant: 1, 2, 5, 10, 24, 90, 770, 941, 11759, 54097, 231973, ..., and they occur at 1, 2, 3, 10, 15, 23, 104, 1701, 2445, 18995, 60037, ... - Robert G. Wilson v, Dec 09 2013

			2.685452001065306445309714835... = 2 + 1/(1 + 1/(2 + 1/(5 + 1/(1 + ...))))
[a_0; a_1, a_2, ...] = [2, 1, 2, ...]

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..999
H. Havermann, Simple Continued Fraction Expansion of Khinchin's Constant
D. Shanks and J. W. Wrench, Jr., Khintchine's constant, Amer. Math. Monthly, 66 (1959), 276-279.
J. W. Wrench, Further evaluation of Khintchine's constant, Math. Comp., 14 (1960), 370-371.
J. W. Wrench, Jr. and D. Shanks, Questions concerning Khintchine's constant and the efficient computation of regular continued fractions, Math. Comp., 20 (1966), 444-448.
G. Xiao, Contfrac
Eric Weisstein's World of Mathematics, Khinchin's Constant Continued Fraction
Index entries for continued fractions for constants

Cf. A002210.

Mathematica
```
ContinuedFraction[Khinchin, 100]
```

More terms from Robert G. Wilson v, Oct 31 2001

A085361 Decimal expansion of the number c = Sum_{n>=1} (zeta(n+1)-1)/n.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jun 25 2003

The Alladi-Grinstead constant (A085291) is exp(c-1).

			0.78853056591150896106027632345455466647274966822328164975515640230178...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 62.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 528 and 538.
Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Vol. 29, No. 2 (June 1988), pp. 196-205.
Eric Weisstein's World of Mathematics, Alladi-Grinstead Constant.
Eric Weisstein's World of Mathematics, Convergence Improvement.

Cf. A002210, A085291, A242624, A244109, A245254.

SetDefaultRealField(RealField(120)); L:=RiemannZeta(); (&+[(Evaluate(L,n+1)-1)/n: n in [1..1000]]); // G. C. Greubel, Nov 15 2018

evalf(sum((Zeta(n+1)-1)/n, n=1..infinity), 120); # Vaclav Kotesovec, Dec 11 2015
evalf(Sum(-(-1)^k*Zeta(1, k), k = 2..infinity), 120); # Vaclav Kotesovec, Jun 18 2021

Sum[(-1+Zeta[1+n])/n,{n,Infinity}]
NSum[Log[k]/(k*(k+1)), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms ->5000, Method -> {NIntegrate, MaxRecursion -> 100}] (* Vaclav Kotesovec, Dec 11 2015 *)

suminf(n=1,(zeta(n+1)-1-2^(-n-1))/n)+log(2)/2 \\ Charles R Greathouse IV, Feb 20 2012

sumalt(k=2, -(-1)^k * zeta'(k)) \\ Vaclav Kotesovec, Jun 17 2021

import mpmath
mpmath.mp.pretty=True; mpmath.mp.dps=108 #precision
mpmath.nsum(lambda n: (-1+mpmath.zeta(1+n))/n, [1,mpmath.inf]) # Peter Luschny, Jul 14 2012

numerical_approx(sum((zeta(k+1)-1)/k for k in [1..1000]), digits=120) # G. C. Greubel, Nov 15 2018

Equals Sum_{n>=2} log(n/(n-1))/n = Sum_{n>=1, k>=2} 1/(n*k^(n+1)). [From Mathworld links]
Equals -Sum_{k>=2} (-1)^k * zeta'(k). - Vaclav Kotesovec, Jun 17 2021
Equals log(A245254) = Sum_{k>=1} log(k)/(k*(k+1)). - Amiram Eldar, Jun 27 2021
Equals -log(A242624). - Amiram Eldar, Feb 06 2022

cp's OEIS Frontend

A064994 A Beatty sequence from Khinchin's constant (A002210).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A064995 A Beatty sequence from Khintchin's constant (A002210).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A100485 Decimal expansion of: (1) a simple-continued-fraction-like nesting in which all "partial quotients" are Khinchin's constant (A002210), or, equivalently, (2) the positive solution p of the polynomial p^2 - Khinchin*p - 1 = 0.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A119928 Continued fraction expansion of the value of Minkowski's question mark function at Khinchin's constant (A002210).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A119929 Decimal expansion of the value of Minkowski's question mark function at Khinchin's constant (A002210).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A100611 Continued fraction expansion of (Khinchin + (4+Khinchin*Khinchin)^(1/2))/2 [A100485], where Khinchin is Khinchin's constant (A002210).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A175819 Partial sums of digits of decimal expansion of Khinchin's constant (sequence A002210).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A131688 Decimal expansion of the constant Sum_{k>=1} log(k + 1) / (k * (k + 1)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

SageMath

Formula

Extensions

A002211 Continued fraction for Khintchine's constant.

Views

Author

Keywords