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A190142 Decimal expansion of limit sqrt(2’*sqrt(3’*sqrt(4’*sqrt(5’*sqrt(6’*...))))), where n’ is the arithmetic derivative of n.

%I A190142 #10 Feb 21 2014 10:14:17
%S A190142 1,2,9,1,3,4,3,1,7,2,1,0,7,2,5,9,5,4,6,1,3,8,8,2,2,9,5,6,5,3,3,4,3,8,
%T A190142 6,9,7,0,3,2,2,0,5,0,2,2,9,6,1,8,7,4,2,4,0,3,6,5,9,8,4,8,6,7,3,9,6,4,
%U A190142 3,9,4,9,0,6,5,2,4,1,4,9,4,0,8,9,9,7,7,9,0,3,7,0,7,9,3,4,1,3,7,3,3,8,0,0,4
%N A190142 Decimal expansion of limit sqrt(2’*sqrt(3’*sqrt(4’*sqrt(5’*sqrt(6’*...))))), where n’ is the arithmetic derivative of n.
%e A190142 1.29134317210725954613882...
%p A190142 with(numtheory);
%p A190142 P:=proc(i)
%p A190142 local a,f,n,p,pfs;
%p A190142 a:=1;
%p A190142 for n from i by -1 to 2 do
%p A190142   pfs:=ifactors(n)[2];
%p A190142   f:=n*add(op(2,p)/op(1,p),p=pfs) ;
%p A190142   a:=f*sqrt(a);
%p A190142 od;
%p A190142 print(evalf(sqrt(a),150));
%p A190142 end:
%p A190142 P(500);
%t A190142 digits = 105; d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; f[n_] := f[n] = Fold[Sqrt[d[#2]*#1]&, n, Range[n-1, 2, -1]] // RealDigits[#, 10, digits]& // First; f[digits]; f[n = 2*digits]; While[f[n] != f[n/2], n = 2*n]; f[n] (* _Jean-François Alcover_, Feb 21 2014 *)
%Y A190142 Cf. A003415, A112302, A171759, A188834, A188835.
%K A190142 nonn,cons
%O A190142 1,2
%A A190142 _Paolo P. Lava_, May 05 2011