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A210937 Decimal expansion of the continued fraction 1'+1/(2'+2/(3'+3/...)), where n' is the arithmetic derivative of n.

%I A210937 #25 Feb 21 2014 04:55:49
%S A210937 4,2,1,4,7,8,1,6,1,2,9,8,8,6,7,3,0,9,0,6,2,0,0,9,1,1,0,4,1,1,2,1,3,6,
%T A210937 4,3,1,1,1,4,6,0,3,3,5,0,7,7,6,8,0,9,0,3,9,6,8,4,3,3,7,4,7,8,7,9,0,8,
%U A210937 7,9,1,4,5,4,0,0,2,2,2,0,4,8,8,1,6,9,0,0,8,5,8,7,0,5,4,9,6,8,4,4,7,5,3,5,8,2,8,2,4,3,0,7,7,2,5,0,5,0,2,4,2,5,4,2,5,8,2,8,2
%N A210937 Decimal expansion of the continued fraction 1'+1/(2'+2/(3'+3/...)), where n' is the arithmetic derivative of n.
%C A210937 A good approximation up to the 9th decimal digit is 4796/11379.
%D A210937 1
%e A210937 0.42147816129886730906200911...
%p A210937 with(numtheory);
%p A210937 A210937:= proc(n)
%p A210937 local a,b,c,I,p,pfs;
%p A210937 b:=1;
%p A210937 for i from n by -1 to 2 do
%p A210937   pfs:=ifactors(i)[2]; a:=i*add(op(2,p)/op(1,p),p=pfs); b:=1/b*a+i;
%p A210937 od;
%p A210937 print(evalf(b,500));
%p A210937 end:
%p A210937 A210937(10000);
%t A210937 digits = 129; d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1&, FactorInteger[n], {1}]]; f[m_] := f[m] = Fold[d[#2]+#2/#1&, 1, Range[m] // Reverse] // RealDigits[#, 10, digits]& // First; f[digits]; f[m = 2digits]; While[f[m] != f[m/2], m = 2m]; f[m] (* _Jean-François Alcover_, Feb 21 2014 *)
%Y A210937 Cf. A003415, A190144, A190145, A190146, A190147, A209873.
%K A210937 nonn,cons
%O A210937 0,1
%A A210937 _Paolo P. Lava_, May 11 2012