A267251 Decimal expansion of Product_{i>=1} (1-1/prime(i))/(1-1/sqrt(prime(i)*prime(i+1))).

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

Offset

0,1

Links

Table of n, a(n) for n=0..16.

Paul Erdős, Solution to Advanced Problem 4413, American Mathematical Monthly, 59 (1952) 259-261.

Example

0.67293388179859770...
From Jon E. Schoenfield, Jan 28 2018: (Start)
Define the partial product y_j = Product_{i=1..PrimePi(j)-1} (1-1/prime(i))/(1-1/sqrt(prime(i)*prime(i+1))); then 2*y_(2^b) - y_(2^(b-1)) converges fairly quickly to lim_{j->infinity} y_j = 0.67293388179859770...:
   b           y_(2^b)            2*y_(2^b) - y_(2^(b-1))
  ==   ========================   ========================
   1   1.0000000000000000000...   ------------------------
   2   0.8449489742783178098...   0.6898979485566356196...
   3   0.7310664129192713972...   0.6171838515602249847...
   4   0.7016018086413063157...   0.6721372043633412342...
   5   0.6843047236120372449...   0.6670076385827681741...
   6   0.6785904879742426949...   0.6728762523364481450...
   7   0.6756179719208981466...   0.6726454558675535982...
   8   0.6742838913222028614...   0.6729498107235075762...
   9   0.6735974784100733488...   0.6729110654979438362...
  10   0.6732641297588515055...   0.6729307811076296623...
  11   0.6730990828541563251...   0.6729340359494611447...
  12   0.6730161366254012027...   0.6729331903966460803...
  13   0.6729749724000593392...   0.6729338081747174757...
  14   0.6729544253323538140...   0.6729338782646482887...
  15   0.6729441568308331961...   0.6729338883293125783...
  16   0.6729390172929284098...   0.6729338777550236236...
  17   0.6729364489209538789...   0.6729338805489793480...
  18   0.6729351653593885893...   0.6729338817978232998...
  19   0.6729345235639937111...   0.6729338817685988329...
  20   0.6729342026805519869...   0.6729338817971102627...
  21   0.6729340422395265924...   0.6729338817985011978...
  22   0.6729339620187032430...   0.6729338817978798937...
  23   0.6729339219086747633...   0.6729338817986462835...
  24   0.6729339018535990721...   0.6729338817985233809...
  25   0.6729338918261069776...   0.6729338817986148831...
  26   0.6729338868123465563...   0.6729338817985861350...
  27   0.6729338843054725858...   0.6729338817985986153...
  28   0.6729338830520350245...   0.6729338817985974632...
  29   0.6729338824253162288...   0.6729338817985974332...
  30   0.6729338821119569733...   0.6729338817985977178...
  31   0.6729338819552773332...   0.6729338817985976930...
  32   0.6729338818769375185...   0.6729338817985977038...
  33   0.6729338818377676111...   0.6729338817985977038...
  34   0.6729338818181826575...   0.6729338817985977039...
(End)

Mathematica

Take[First@ RealDigits@ N[Product[(1 - 1/Prime@ i)/(1 - 1/Sqrt[Prime[i] Prime[i + 1]]), {i, 100000}]], 5] (* Michael De Vlieger, Jan 12 2016 *)

Crossrefs

Cf. A245630, A245636.

Sequence in context: A153971 A308039 A316165 * A259526 A108664 A160155

Adjacent sequences: A267248 A267249 A267250 * A267252 A267253 A267254

Keyword

nonn,cons,more

Author

Michel Marcus, Jan 12 2016

Extensions

Three more digits from Jean-François Alcover, Jan 13 2016

Nine more digits from Jon E. Schoenfield, Jan 28 2018

Status

approved