login
This site is supported by donations to The OEIS Foundation.

 

Logo

Invitation: celebrating 50 years of OEIS, 250000 sequences, and Sloane's 75th, there will be a conference at DIMACS, Rutgers, Oct 9-10 2014.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A037077 Decimal expansion of upper limit of - 1^(1/1) + 2^(1/2) - 3^(1/3) + ... 6
1, 8, 7, 8, 5, 9, 6, 4, 2, 4, 6, 2, 0, 6, 7, 1, 2, 0, 2, 4, 8, 5, 1, 7, 9, 3, 4, 0, 5, 4, 2, 7, 3, 2, 3, 0, 0, 5, 5, 9, 0, 3, 0, 9, 4, 9, 0, 0, 1, 3, 8, 7, 8, 6, 1, 7, 2, 0, 0, 4, 6, 8, 4, 0, 8, 9, 4, 7, 7, 2, 3, 1, 5, 6, 4, 6, 6, 0, 2, 1, 3, 7, 0, 3, 2, 9, 6, 6, 5, 4, 4, 3, 3, 1, 0, 7, 4, 9, 6, 9, 0, 3, 8, 4, 2 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From Daniel Forgues, Apr 20 2011: (Start)

The series Sum_(n=1..infty) (-1)^n n^(1/n) diverges (oscillates) with the upper limit given by this sequence and the lower limit being the upper limit - 1.

The series Sum_(n=1..infty) (-1)^n (n^(1/n)-1) converges to this upper limit. (End)

Added a program that Richard Crandall wrote (and I tweaked a little). See the link below, sections 7.5 and 7.6. - Marvin Ray Burns Feb 19 2013

In Mathematica Program 3 "pr = Floor[1.005 pre]}, chunksize = cores*tsize;" changed back to "pr = Floor[1.02 pre]}, chunksize = cores*tsize;" per Crandall's original code. - Marvin Ray Burns, Dec 05 2013

3014991 terms of this sequence are known. - Marvin Ray Burns, Sept 27 2014

REFERENCES

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

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 0..1000

Marvin Ray Burns, Mathematica Notebook of first known 314159 digit computation finished on Sep 04 2012

Marvin Ray Burns, Text version of 314159 digits

Marvin Ray Burns, Mathematica Notebook of first known 3014991 digit computation finished on Sep 21 2014

Richard E. Crandall, Unified algorithms for polylogarithm, L-series, and zeta variants (53 pages)

OEIS Wiki, MRB constant

Simon Plouffe Tables of Constants [From Simon Plouffe]

Eric Weisstein's World of Mathematics, MRB Constant

Eric Weisstein's World of Mathematics, Power Tower

EXAMPLE

=0.1878596424620671202485179340542732300559030949001387861720046840894772315...

MAPLE

A037077 := proc (e) local a, b, c, d, s, k, n, m; if e < 100 then n := 31+e; Digits := 31+e else n := 131*round((1/100)*e); Digits := 131*round((1/100)*e) end if; a := array(0 .. n-1); a[0] := 1; for m to n-1 do a[m] := ((1/2)*sinh(2*ln(m+1)/(m+1))+cosh(ln(m+1)/(m+1))^2-1)/sinh(ln(m+1)/(m+1)) end do; d := (1/2)*(3+2*2^(1/2))^n+(1/2)/(3+2*2^(1/2))^n; b := -1; c := -d; s := 0; for k from 0 to n-1 do c := b-c; b := 2*b*(k^2-n^2)/((2*k+1)*(k+1)); s := s+c*a[k] end do; Digits := e; print(evalf(1/2-s/d)) end proc; A037077(1000) { where 1000 is the number of digits desired }

MATHEMATICA

Program 1

f[mx_] := Block[{$MaxExtraPrecision = mx + 8, a, b = -1, c = -1 - d, d = (3 + Sqrt[8])^n, n = 131 Ceiling[mx/100], s = 0}, a[0] = 1; d = (d + 1/d)/2; For[m = 1, m < n, a[m] = (1 + m)^(1/(1 + m)); m++]; For[k = 0, k < n, c = b - c; b = b (k + n) (k - n)/((k + 1/2) (k + 1)); s = s + c*a[k]; k++]; N[1/2 - s/d, mx]]; RealDigits[ f[105], 10][[1]] (* mx is the number of digits desired  - Marvin Ray Burns, Aug 05 2007*)

Program 2

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

Program 3

(* Fastest (at MRB's end) as of Jan 06 2013. For use with large calculations (5, 000-3, 000, 000 digits) *)

prec = 5000; (* Number of required decimals. *) ClearSystemCache[];

T0 = SessionTime[];

expM[pre_] :=

  Module[{a, d, s, k, bb, c, n, end, iprec, xvals, x, pc, cores = 6,

    tsize = 2^7, chunksize, start = 1, ll, ctab,

    pr = Floor[1.02 pre]}, chunksize = cores*tsize;

   n = Floor[1.32 pr];

   end = Ceiling[n/chunksize];

   Print["Iterations required: ", n];

   Print["end ", end];

   Print[end*chunksize]; d = ChebyshevT[n, 3];

   {b, c, s} = {SetPrecision[-1, 1.1*n], -d, 0};

   iprec = Ceiling[pr/27];

   Do[xvals = Flatten[ParallelTable[Table[ll = start + j*tsize + l;

        x = N[E^(Log[ll]/(ll)), iprec];

        pc = iprec;

        While[pc < pr, pc = Min[3 pc, pr];

         x = SetPrecision[x, pc];

         y = x^ll - ll;

         x = x (1 - 2 y/((ll + 1) y + 2 ll ll)); ]; (*N[Exp[Log[ll]/ll],

        pr]*)x, {l, 0, tsize - 1}], {j, 0, cores - 1},

       Method -> "CoarsestGrained"]];

    ctab = ParallelTable[Table[c = b - c;

       ll = start + l - 2;

       b *= 2 (ll + n) (ll - n)/((ll + 1) (2 ll + 1));

       c, {l, chunksize}], Method -> "CoarsestGrained"];

    s += ctab.(xvals - 1);

    start += chunksize;

    Print["done iter ", k*chunksize, " ", SessionTime[] - T0]; , {k, 0,

      end - 1}];

   N[-s/d, pr]];

t2 = Timing[MRBtest2 = expM[prec]; ]; Print[MRBtest2] (* Richard Crandall via Marvin Ray Burns, Feb 19 2013 *)

PROG

(PARI) sumalt(x=1, (-1)^x*((x^(1/x))-1))

CROSSREFS

Cf. A052110, A157852, A160755, A173273.

Sequence in context: A180311 A103984 A203914 * A094106 A242968 A021536

Adjacent sequences:  A037074 A037075 A037076 * A037078 A037079 A037080

KEYWORD

cons,nonn,changed

AUTHOR

Marvin Ray Burns. Entry updated Jan 30 2009, Jun 21 2009, Dec 11 2009, Sep 04 2010, Jun 23 2011, Sep 08 2012

EXTENSIONS

Definition corrected by Daniel Forgues, Apr 20 2011

Broken links replaced by Marvin Ray Burns, Sep 28 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified September 30 10:09 EDT 2014. Contains 247419 sequences.