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%I #24 Jan 22 2022 08:46:28
%S 6,7,2,9,3,3,8,8,1,7,9,8,5,9,7,7,0
%N Decimal expansion of Product_{i>=1} (1-1/prime(i))/(1-1/sqrt(prime(i)*prime(i+1))).
%H Paul Erdős, <a href="http://www.jstor.org/stable/2306529">Solution to Advanced Problem 4413</a>, American Mathematical Monthly, 59 (1952) 259-261.
%e 0.67293388179859770...
%e From _Jon E. Schoenfield_, Jan 28 2018: (Start)
%e 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...:
%e b y_(2^b) 2*y_(2^b) - y_(2^(b-1))
%e == ======================== ========================
%e 1 1.0000000000000000000... ------------------------
%e 2 0.8449489742783178098... 0.6898979485566356196...
%e 3 0.7310664129192713972... 0.6171838515602249847...
%e 4 0.7016018086413063157... 0.6721372043633412342...
%e 5 0.6843047236120372449... 0.6670076385827681741...
%e 6 0.6785904879742426949... 0.6728762523364481450...
%e 7 0.6756179719208981466... 0.6726454558675535982...
%e 8 0.6742838913222028614... 0.6729498107235075762...
%e 9 0.6735974784100733488... 0.6729110654979438362...
%e 10 0.6732641297588515055... 0.6729307811076296623...
%e 11 0.6730990828541563251... 0.6729340359494611447...
%e 12 0.6730161366254012027... 0.6729331903966460803...
%e 13 0.6729749724000593392... 0.6729338081747174757...
%e 14 0.6729544253323538140... 0.6729338782646482887...
%e 15 0.6729441568308331961... 0.6729338883293125783...
%e 16 0.6729390172929284098... 0.6729338777550236236...
%e 17 0.6729364489209538789... 0.6729338805489793480...
%e 18 0.6729351653593885893... 0.6729338817978232998...
%e 19 0.6729345235639937111... 0.6729338817685988329...
%e 20 0.6729342026805519869... 0.6729338817971102627...
%e 21 0.6729340422395265924... 0.6729338817985011978...
%e 22 0.6729339620187032430... 0.6729338817978798937...
%e 23 0.6729339219086747633... 0.6729338817986462835...
%e 24 0.6729339018535990721... 0.6729338817985233809...
%e 25 0.6729338918261069776... 0.6729338817986148831...
%e 26 0.6729338868123465563... 0.6729338817985861350...
%e 27 0.6729338843054725858... 0.6729338817985986153...
%e 28 0.6729338830520350245... 0.6729338817985974632...
%e 29 0.6729338824253162288... 0.6729338817985974332...
%e 30 0.6729338821119569733... 0.6729338817985977178...
%e 31 0.6729338819552773332... 0.6729338817985976930...
%e 32 0.6729338818769375185... 0.6729338817985977038...
%e 33 0.6729338818377676111... 0.6729338817985977038...
%e 34 0.6729338818181826575... 0.6729338817985977039...
%e (End)
%t 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 *)
%Y Cf. A245630, A245636.
%K nonn,cons,more
%O 0,1
%A _Michel Marcus_, Jan 12 2016
%E Three more digits from _Jean-François Alcover_, Jan 13 2016
%E Nine more digits from _Jon E. Schoenfield_, Jan 28 2018