A037077 -id:A037077 - cp's OEIS frontend

A052110 Decimal expansion of c^c^c^... where c is the constant defined in A037077.

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

Offset: 0

Marvin Ray Burns Jan 20 2000, Mar 28 2008, Nov 08 2009, Mar 24 2010, Jun 27 2011

See (Weisstein) link on Power Tower.

			0.4619214401644114454085886426141945786350282801364882284434162927358917250...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 448-452.

Eric Weisstein's World of Mathematics, Power Tower.
Gus Wiseman, Tetration.
OEIS Wiki, Tetration.
OEIS Wiki, MRB constant.
Wikipedia, Tetration.

Cf. A037077, A000027, A000312, A002488, A073230.

n = 105; M = NSum[(-1)^n*(n^(1/n) - 1), {n, 1, Infinity}, WorkingPrecision -> n + 10, Method -> "AlternatingSigns"]; L = Log[M]; N[-ProductLog[-L]/L, n] (* Marvin Ray Burns, Mar 08 2013 *)

default(realprecision,66);
M=sumalt(x=1,(-1)^x*((x^(1/x))-1));
solve(x=.46,.462,x^(1/x)-M)

Simplified definition by Marvin Ray Burns, Mar 08 2013

A173273 Decimal expansion of 1 - 2*A037077.

Original entry on oeis.org

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

Offset: 0

Marvin Ray Burns, Feb 14 2010, Feb 24 2010, Mar 05 2010

			0.624280715...

Cf. A037077.

digits = 105; 1-2*NSum[ (-1)^n*((n^(1/n)) - 1), {n, 1, Infinity}, WorkingPrecision -> digits+10, Method -> "AlternatingSigns"] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 15 2013 *)

A248660 Simple continued fraction expansion of the constant defined in A037077.

Original entry on oeis.org

0, 5, 3, 10, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 12, 2, 17, 2, 2, 1, 1, 17, 1, 6, 4, 1, 3, 3, 4, 2, 1, 262, 2, 1, 4, 1, 49, 2, 1, 9, 1, 2, 1, 1, 4, 23, 26, 6, 6, 5, 3, 3, 1, 1, 1, 144, 9, 1, 1, 5, 1, 3, 1, 1, 5, 13, 8619, 2, 1, 45, 2, 1, 1, 2, 1, 4, 5, 1, 7, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 6, 3, 1, 1, 2, 2, 7, 3, 136, 1

Offset: 0

Marvin Ray Burns, Jan 11 2015

A037077 is sometimes called the MRB constant.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 450.

Eric Weisstein's World of Mathematics, MRB Constant

Cf. A037077 (decimal expansion).

MRB:= Sum((-1)^k*(k^(1/k)-1),k=1..infinity):
V:= evalf[150](MRB):
subs(`...`=NULL, numtheory:-cfrac(V, 100, 'quotients')); # Robert Israel, Jan 12 2015

m = NSum[(-1)^n*(n^(1/n) - 1), {n, 1, Infinity},
   WorkingPrecision -> 100, Method -> "AlternatingSigns"];
ContinuedFraction[m]

contfrac(sumalt(x=1, (-1)^x*((x^(1/x))-1)) ) \\ Michel Marcus, Jan 12 2015

A157852 Decimal expansion of the absolute value of lim_{N -> infinity} Integral_{x=1..2N} e^(iPix)x^(1/x).

Original entry on oeis.org

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

Offset: 0

Marvin Ray Burns, Mar 07 2009

This constant is the integral analog of the constant described in A037077 since e^(i*Pi*x) =(-1)^x. While A037077 was named the MRB constant by Simon Plouffe, Marvin Ray Burns named this constant MKB after his wife at the time.
This constant is hard to integrate and very slow to converge, so it takes a combination of modern methods to calculate many digits!
This constant could be written as a special value, for omega=Pi, of the function f(omega) = lim_{N->infinity} Integral_{x = Pi/omega .. 2N8(Pi/omega)} (exp(i*omega*x)*x^(1/x)), a kind of discretely sampled Fourier transform of x^(1/x). This stresses the fact that it is a complex entity. People who desire to underline the similarity of this integral to the MRB alternating series (A037077) often write the factor exp(i*Pi*x) as (-1)^x, which can be a bit confusing because it hides the imaginary unit. - Stanislav Sykora, Apr 08 2016

			After integrating from 1 to 15 million the absolute value of the integral is approximately 0.687652_7, after integrating from 1 to 20 million approximately 0.687652_6.

Marvin Ray Burns, Table of n, a(n) for n = 0..19999
Marvin Ray Burns, Author's public inquiry 1
Marvin Ray Burns, Author's public inquiry 2
Marvin Ray Burns, How I calculated the digits of the MKB constant
Marvin Ray Burns, Paper on 20000 digits in a Mathematica notebook (Digits checked by different formula, computing more digits)
Marvin Ray Burns, How 40,000 digits of A157852 was successfully computed and checked to 35,000 digits
Marvin Ray Burns, Mathematica Notebook of 54,386 digits of A157852 computed with increasing MaxRecursion, with proof at bottom.
Marvin Ray Burns, How_I_found_A157852_sum.
Marvin Ray Burns, A157852 sum PDFK
R. J. Mathar, Numerical evaluation of the oscillatory integral over exp(i*pi*x)x^(1/x) between 1 and infinity, arxiv:0912.3844 [math.CA], 2009-2010.

Integrating A037077 instead of summing.

Cf. A037077, A255727 (real part), A255728 (imaginary part).

# After Marvin Ray Burns's "Program 3".
f := (n, x) -> seq(x, 0..n):
m := n -> (Pi/I)^n * MeijerG([[], [f(n, 1)]], [[1-n, f(n, 0)], []], -I*Pi):
s := n -> abs(add(m(k), k = 1..n) - 2)/Pi:
# s(n) approximates the constant for n -> oo and suitable chosen precision.
seq(evalf(s(n), 22), n = 1..3); # Peter Luschny, Nov 16 2021

a = NIntegrate[ x^(1/x)*Cos[Pi*x], {x, 1, 10^20}, WorkingPrecision -> 30, MaxRecursion -> 70]; b = NIntegrate[ x^(1/x)*Sin[Pi*x], {x, 1, 10^20}, WorkingPrecision -> 30, MaxRecursion -> 70]; RealDigits[ Sqrt[a^2 + b^2], 10, 18] // First (* Jean-François Alcover, Feb 14 2013 *)
(* Program 2: to compute and verify 1000s of digits through a different formula. *)
g[x_] = x^(1/x); t = (Timing[
    MKB = -(I NIntegrate[(g[(1 + t I)]) ( Exp[-Pi t]), {t, 0,
           Infinity}, WorkingPrecision -> 2410,
          Method -> "Trapezoidal", MaxRecursion -> 10] + I/Pi)])[[
  1]]; Print["Timing for calculation=", t]; t = (Timing[
    MKB1 = (1/Pi  NIntegrate[
         g'[1 + I t] Exp[-Pi t], {t, 0, Infinity},
         WorkingPrecision -> 2410, Method -> "Trapezoidal",
         MaxRecursion -> 10] - 2 I/Pi)])[[
  1]]; Print["Timing for verification=", t]; err =
test - test2; Print["Error=", N[err, 20]];Abs[MKB] (* MaxRecursion -> 13 works for 10,000 digits. Marvin Ray Burns, Apr 18 2021 *)
(* Program 3: An infinite sum involving the Meijer G function. Compare the discussion near the end of "How I calculated the digits of the MKB constant" and all of Cloud Notebook "How_I_found_A157852_sum" in the link section. *)
f[n_] := MeijerG[{{},Table[1, {n+1}]}, {Prepend[Table[0, n+1], -n + 1], {}}, -I Pi];
Abs[Sum[(I/Pi)^(1 - n) N[f[n], 22], {n, 1, 15}] - 2 I/Pi] (* Marvin Ray Burns, Nov 15 2021 *)

Equals sqrt(A255727^2 + A255728^2). - Joerg Arndt, Apr 05 2016

Edited by N. J. A. Sloane, Mar 13 2009
Corrected and edited by Marvin Ray Burns, Apr 03 2009
8 more digits from R. J. Mathar, Nov 30 2009, 3 more Jan 03 2011, 3 more on Feb 25 2013
15 more digits added by Marvin Ray Burns, Feb 26 2013
Many more digits added by Marvin Ray Burns, May 11 2015
Edited by Marvin Ray Burns, Aug 06 2015
Edited by Marvin Ray Burns, Jun 18 2017

A160755 Number of correct digits of the MRB constant derived from the sequence of partial sums up to m=10^n terms as defined by S[n]= Sum[(-1)^k*(k^(1/k)-1),{k,m}].

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49

Offset: 1

Marvin Ray Burns, May 25 2009

Adding the series -1+sqrt(2)-3^(1/3)+4^(1/4)..., according to this sequence, 10 billion terms must be added to arrive at 11 accurate digits of the MRB constant.

			For n=1, a(n)=1 because after 10^1 partial sums of -1+sqrt(2)-3^(1/3)+4^(1/4)... you get one accurate digit of the MRB constant.
For n=2, a(n)=2 because after 10^2 partial sums you get two accurate digits and so on.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 450. ISBN 0521818052.

Henri Cohen, Fernando Rodriguez Villegas and Don Zagier, Convergence Acceleration of Alternating Series, Experimental Mathematics, 9:1 (2000).
Eric Weisstein's World of Mathematics, MRB Constant.
Wikipedia, Mathematical constant

Cf. A037077 (the MRB constant).

m = NSum[(-1)^n*(n^(1/n) - 1), {n, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 1000]; Table[-Floor[Log[10, Abs[m - NSum[(-1)^n*(n^(1/n) - 1), {n, 10^a}, Method ->"AlternatingSigns", WorkingPrecision -> 1000]]]], {a,1, 50}]

Corrections from Marvin Ray Burns, Jun 05 2009
Link to Wikipedia replaced by up-to-date version; keyword:less added R. J. Mathar, Aug 04 2010
Corrections by Marvin Ray Burns, Aug 21 2010, Jul 15 2012

A177218 Decimal expansion of the integral over cos(Pix)x^(1/x) between 1/e and e.

Original entry on oeis.org

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

Offset: 0

Marvin Ray Burns, May 04 2010

Strangely close to A037077 which is a sum of the integrand from 1 to infinity.

			0.187779...

Marvin Ray Burns, Author's original inquiry
R. J. Mathar, Numerical evaluation of the oscillatory integral over exp(i*pi*x)x^(1/x) between 1 and infinity, arXiv:0912.3844

A157852 is the same integral from 1 to infinity.

Int( cos(Pi*x)*x^(1/x),x=exp(-1)..exp(1)) ; evalf(%) ; # R. J. Mathar, May 07 2010

RealDigits[ Re[NIntegrate[(-1)^n*n^(1/n), {n, 1/E, E}, WorkingPrecision -> 200]]]

Definition simplified, keyword:cons inserted, offset corrected by R. J. Mathar, May 07 2010

A250091 Decimal expansion of 1+2^(1/2)+3^(1/3)+4^(1/4)+5^(1/5)+6^(1/6)+7^(1/7).

Original entry on oeis.org

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

Offset: 1

Vincenzo Librandi, Nov 12 2014

Cf. A037077.

1+2*2^(1/2)+3^(1/3)+5^(1/5)+6^(1/6)+7^(1/7);

RealDigits[1 + 2 2^(1/2) + 3^(1/3) + 5^(1/5) + 6^(1/6) + 7^(1/7), 10, 120][[1]]

sum(n=1,7,sqrtn(n,n)) \\ Charles R Greathouse IV, Apr 20 2016

Equals 9.3188817588682147714986428688188651966599608644706733821414765...

cp's OEIS Frontend

A052110 Decimal expansion of c^c^c^... where c is the constant defined in A037077.

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A173273 Decimal expansion of 1 - 2*A037077.

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A248660 Simple continued fraction expansion of the constant defined in A037077.

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A157852 Decimal expansion of the absolute value of lim_{N -> infinity} Integral_{x=1..2*N} e^(i*Pi*x)*x^(1/x).

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A160755 Number of correct digits of the MRB constant derived from the sequence of partial sums up to m=10^n terms as defined by S[n]= Sum[(-1)^k*(k^(1/k)-1),{k,m}].

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A177218 Decimal expansion of the integral over cos(Pi*x)*x^(1/x) between 1/e and e.

Views

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Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A250091 Decimal expansion of 1+2^(1/2)+3^(1/3)+4^(1/4)+5^(1/5)+6^(1/6)+7^(1/7).

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Magma

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A157852 Decimal expansion of the absolute value of lim_{N -> infinity} Integral_{x=1..2N} e^(iPix)x^(1/x).

A177218 Decimal expansion of the integral over cos(Pix)x^(1/x) between 1/e and e.