A087498 -id:A087498 - cp's OEIS frontend

A087491 Decimal expansion of the Khinchin harmonic mean K_{-1}.

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

Offset: 1

Eric W. Weisstein, Sep 09 2003

Khinchin's constant is K_0 (A002210).

			1.7454056624073468634945963096836610672949366187...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.8, p. 61.

Eric Weisstein's World of Mathematics, Khinchin's Constant.
Eric Weisstein's World of Mathematics, Khinchin Harmonic Mean.
Wikipedia, Khinchin's constant.

Cf. A002210, A087491, A087492, A087493, A087494, A087495, A087496, A087497, A087498, A087499, A087500.

digits = 102; exactEnd = 1000; f[n_] = (1 - 1/(n + 1)^2)^(-1/n); s[n_] = Series[Log[f[n]], {n, Infinity, digits}] // Normal // N[#, digits] &; exactSum = Sum[Log[f[n]], {n, 1, exactEnd}] // N[#, digits] &; extraSum = Sum[s[n], {n, exactEnd + 1, Infinity}] // N[#, digits] &; A087491 = Log[2]/(exactSum + extraSum) // RealDigits // First  (* Jean-François Alcover, Feb 06 2013 *)
RealDigits[Log[2]/NSum[Log[(1 - 1/(n + 1)^2)^(-1/n)], {n, Infinity}, NSumTerms -> 10^4, WorkingPrecision -> 250, PrecisionGoal -> 110]][[1, ;; 100]] (* Eric W. Weisstein, Dec 10 2017 *)

Equals (Sum_{n>=1} -log2(1 - 1/(n+1)^2) * n^(-1))^(-1). - Jianing Song, Aug 08 2021

A087500 Decimal expansion of Khinchin mean K_{-10}.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Sep 09 2003

Khinchin's constant is K_0.

			1.09187704...

Eric Weisstein's World of Mathematics, Khinchin's Constant

Cf. A002210, A087491, A087492, A087493, A087494, A087495, A087496, A087497, A087498, A087499, A087500.

m = 10; digits = 100; exactEnd = 1000; f[n_] = -(Log[1 - (1 + n)^(-2)]/(n^m*Log[2])); s[n_] = Series[f[n], {n, Infinity, digits}] // Normal // N[#, digits]&; exactSum = Sum[f[n], {n, 1, exactEnd}] // N[#, digits]&; extraSum = Sum[s[n], {n, exactEnd + 1, Infinity}] // N[#, digits]&; (exactSum + extraSum)^(-1/m) // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 14 2013 *)

More terms from Jean-François Alcover, Feb 14 2013

A087492 Decimal expansion of Khinchin mean K_{-2}.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Sep 09 2003

Khinchin's constant is K_0.

			1.4503403284956304060529830766806978814082999796059041821717...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.8, p. 61.

Eric Weisstein's World of Mathematics, Khinchin's Constant.

Cf. A002210, A087491, A087492, A087493, A087494, A087495, A087496, A087497, A087498, A087499, A087500.

digits = 102; exactEnd = 1000; f[n_] = -(Log[1 - (1 + n)^(-2)]/(n^2*Log[2])); s[n_] = Series[f[n] , {n, Infinity, digits}] // Normal // N[#, digits]&; exactSum = Sum[f[n] , {n, 1, exactEnd}] // N[#, digits]&; extraSum = Sum[s[n] , {n, exactEnd + 1, Infinity}] // N[#, digits]&; A087492 = 1/Sqrt[exactSum + extraSum] // RealDigits // First (* Jean-François Alcover, Feb 06 2013 *)

More terms from Jean-François Alcover, Feb 06 2013

A087493 Decimal expansion of Khinchin mean K_{-3}.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Sep 09 2003

Khinchin's constant is K_0.

			1.3135070786879857667173394470727868281581298614...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.8, p. 61.

Eric Weisstein's World of Mathematics, Khinchin's Constant.

Cf. A002210, A087491, A087492, A087493, A087494, A087495, A087496, A087497, A087498, A087499, A087500.

digits = 102; exactEnd = 1000; f[n_] = -(Log[1 - (1 + n)^(-2)]/(n^3*Log[2])); s[n_] = Series[f[n] , {n, Infinity, digits}] // Normal // N[#, digits]&; exactSum = Sum[f[n] , {n, 1, exactEnd}] // N[#, digits]&; extraSum = Sum[s[n] , {n, exactEnd + 1, Infinity}] // N[#, digits]&; A087493 = (exactSum + extraSum)^(-1/3) // RealDigits // First (* Jean-François Alcover, Feb 06 2013 *)

More terms from Jean-François Alcover, Feb 06 2013

A087494 Decimal expansion of Khinchin mean K_{-4}.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Sep 09 2003

Khinchin's constant is K_0.

			1.23696180...

Eric Weisstein's World of Mathematics, Khinchin's Constant

Cf. A002210, A087491, A087492, A087493, A087494, A087495, A087496, A087497, A087498, A087499, A087500.

m = 4; digits = 100; exactEnd = 1000; f[n_] = -(Log[1 - (1 + n)^(-2)]/(n^m*Log[2])); s[n_] = Series[f[n], {n, Infinity, digits}] // Normal // N[#, digits]&; exactSum = Sum[f[n], {n, 1, exactEnd}] // N[#, digits]&; extraSum = Sum[s[n], {n, exactEnd + 1, Infinity}] // N[#, digits]&; (exactSum + extraSum)^(-1/m) // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 14 2013 *)

More terms from Jean-François Alcover, Feb 14 2013

A087495 Decimal expansion of Khinchin mean K_{-5}.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Sep 09 2003

Khinchin's constant is K_0.

			1.18900392...

Eric Weisstein's World of Mathematics, Khinchin's Constant

Cf. A002210, A087491, A087492, A087493, A087494, A087495, A087496, A087497, A087498, A087499, A087500.

m = 5; digits = 100; exactEnd = 1000; f[n_] = -(Log[1 - (1 + n)^(-2)]/(n^m*Log[2])); s[n_] = Series[f[n], {n, Infinity, digits}] // Normal // N[#, digits]&; exactSum = Sum[f[n], {n, 1, exactEnd}] // N[#, digits]&; extraSum = Sum[s[n], {n, exactEnd + 1, Infinity}] // N[#, digits]&; (exactSum + extraSum)^(-1/m) // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 14 2013 *)

A087496 Decimal expansion of Khinchin mean K_{-6}.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Sep 09 2003

Khinchin's constant is K_0.

			1.15655237...

Eric Weisstein's World of Mathematics, Khinchin's Constant

Cf. A002210, A087491, A087492, A087493, A087494, A087495, A087496, A087497, A087498, A087499, A087500.

m = 6; digits = 100; exactEnd = 1000; f[n_] = -(Log[1 - (1 + n)^(-2)]/(n^m*Log[2])); s[n_] = Series[f[n], {n, Infinity, digits}] // Normal // N[#, digits]&; exactSum = Sum[f[n], {n, 1, exactEnd}] // N[#, digits]&; extraSum = Sum[s[n], {n, exactEnd + 1, Infinity}] // N[#, digits]&; (exactSum + extraSum)^(-1/m) // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 14 2013 *)

More terms from Charles R Greathouse IV, Jul 16 2007
Edited by Charles R Greathouse IV, Oct 28 2009
More terms from Jean-François Alcover, Feb 14 2013

A087497 Decimal expansion of Khinchin mean K_{-7}.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Sep 09 2003

Khinchin's constant is K_0.

			1.13332336...

Eric Weisstein's World of Mathematics, Khinchin's Constant

Cf. A002210, A087491, A087492, A087493, A087494, A087495, A087496, A087497, A087498, A087499, A087500.

m = 7; digits = 100; exactEnd = 1000; f[n_] = -(Log[1 - (1 + n)^(-2)]/(n^m*Log[2])); s[n_] = Series[f[n], {n, Infinity, digits}] // Normal // N[#, digits]&; exactSum = Sum[f[n], {n, 1, exactEnd}] // N[#, digits]&; extraSum = Sum[s[n], {n, exactEnd + 1, Infinity}] // N[#, digits]&; (exactSum + extraSum)^(-1/m) // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 14 2013 *)

More terms from Jean-François Alcover, Feb 14 2013

A087499 Decimal expansion of Khinchin mean K_{-9}.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Sep 09 2003

Khinchin's constant is K_0.

			1.10254313...

Eric Weisstein's World of Mathematics, Khinchin's Constant

Cf. A002210, A087491, A087492, A087493, A087494, A087495, A087496, A087497, A087498, A087499, A087500.

m = 9; digits = 100; exactEnd = 1000; f[n_] = -(Log[1 - (1 + n)^(-2)]/(n^m*Log[2])); s[n_] = Series[f[n], {n, Infinity, digits}] // Normal // N[#, digits]&; exactSum = Sum[f[n], {n, 1, exactEnd}] // N[#, digits]&; extraSum = Sum[s[n], {n, exactEnd + 1, Infinity}] // N[#, digits]&; (exactSum + extraSum)^(-1/m) // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 14 2013 *)

More terms from Jean-François Alcover, Feb 14 2013

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A087491 Decimal expansion of the Khinchin harmonic mean K_{-1}.

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A087500 Decimal expansion of Khinchin mean K_{-10}.

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A087492 Decimal expansion of Khinchin mean K_{-2}.

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A087493 Decimal expansion of Khinchin mean K_{-3}.

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A087494 Decimal expansion of Khinchin mean K_{-4}.

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A087495 Decimal expansion of Khinchin mean K_{-5}.

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A087496 Decimal expansion of Khinchin mean K_{-6}.

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