A037077 Decimal expansion of upper limit of - 1^(1/1) + 2^(1/2) - 3^(1/3) + ... .
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
Offset: 0
Examples
0.1878596424620671202485179340542732300559030949001387861720046840894772315...
References
- Steven 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.
- Marvin Ray Burns, Mathematica Notebook of first known 6029991 digit computation of A037077, finished on Mar 30 2021.
- Richard E. Crandall, Unified algorithms for polylogarithm, L-series, and zeta variants (53 pages).
- OEIS Wiki, MRB constant
- Simon Plouffe, From tables of Constants [Original documentation written by M. R. Burns in 1999]
- Eric Weisstein's World of Mathematics, MRB Constant
- Eric Weisstein's World of Mathematics, Power Tower
Programs
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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 }
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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-François Alcover, Feb 15 2013 *) (* Program 3 *) (* Fastest as of Jan 06 2013. For use with large calculations (5,000-3,000,000 digits) *) prec = 5000; (* Number of required digits. *) 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 *) -
PARI
sumalt(x=1,(-1)^x*((x^(1/x))-1))
Extensions
Definition corrected by Daniel Forgues, Apr 20 2011
Comments