|
%I #26 Feb 20 2023 15:04:53
%S 5,4,6,6,8,0,3,2,4,4,6,6,4,2,3,3,7,1,4,6,2,1,7,2,7,2,4,7,0,9,5,0,0,2,
%T 7,2,2,6,3,0,2,5,6,8,9,8,5,2,7,9,2,1,6,6,1,4,5,4,3,4,9,8,1,2,6,4,9,0,
%U 4,2,0,4,8,4,0,8,5,9,4,9,0,0,9,7,3,2,6,7,0,6,5,1,6,0,3,7,3,0,3,0,4,0,6,8,2,1,6,0,7,7,4,4,8,4,2,9,0,5,8
%N Decimal expansion of continued fraction 1'+1/(2'+1/(3'+1/(4'+1/(5'...)))), where n' is the arithmetic derivative of n.
%C Good approximation up to the 9th decimal digit with 22948/41977.
%H G. C. Greubel, <a href="/A190143/b190143.txt">Table of n, a(n) for n = 0..10000</a>
%e 0.546680324466423...
%p with(numtheory);
%p P:=proc(i)
%p local a,f,n,p,pfs;
%p a:=0;
%p for n from i by -1 to 1 do
%p pfs:=ifactors(n)[2];
%p f:=n*add(op(2,p)/op(1,p),p=pfs);
%p a:=1/(a+f);
%p od;
%p print(evalf(1/a,1000));
%p end:
%p P(10000);
%t digits = 119; d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; f[m_] := f[m] = Fold[d[#2] + 1/#1 &, 1, Range[m] // Reverse] // RealDigits[#, 10, digits] & // First; f[digits]; f[m = 2 digits]; While[f[m] != f[m/2], m = 2 m]; f[m] (* _Jean-François Alcover_, Feb 21 2014 *)
%Y Cf. A003415.
%K nonn,cons
%O 0,1
%A _Paolo P. Lava_, May 05 2011
|
