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

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, 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, 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

Offset: 1

Author

Paolo P. Lava, May 05 2011

Keywords

• nonn
• cons

Examples

			1.29134317210725954613882...
		

Crossrefs

Cf. A003415, A112302, A171759, A188834, A188835.

Programs

Maple

with(numtheory);
P:=proc(i)
local a,f,n,p,pfs;
a:=1;
for n from i by -1 to 2 do
  pfs:=ifactors(n)[2];
  f:=n*add(op(2,p)/op(1,p),p=pfs) ;
  a:=f*sqrt(a);
od;
print(evalf(sqrt(a),150));
end:
P(500);

Mathematica

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 *)