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

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, 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, 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

Offset: 0

Author

Paolo P. Lava, May 11 2012

Keywords

• nonn
• cons

Comments

A good approximation up to the 9th decimal digit is 4796/11379.

Examples

			0.42147816129886730906200911...
		

References

• 1

Crossrefs

Cf. A003415, A190144, A190145, A190146, A190147, A209873.

Programs

Maple

with(numtheory);
A210937:= proc(n)
local a,b,c,I,p,pfs;
b:=1;
for i from n by -1 to 2 do
  pfs:=ifactors(i)[2]; a:=i*add(op(2,p)/op(1,p),p=pfs); b:=1/b*a+i;
od;
print(evalf(b,500));
end:
A210937(10000);

Mathematica

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 *)