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%I #15 Jan 19 2019 15:53:40
%S 0,0,3,4,7,2,8,2,5,0,4,3,3,8,6,7,0,8,1,4,7,9,1,7,6,7,3,4,2,4,6,2,3,0,
%T 5,2,7,2,7,3,7,4,5,2,4,3,1,4,7,8,0,7,4,0,5,5,1,1,2,3,8,1,4,1,5,8,4,0,
%U 3,6,9,6,8,5,5,8,2,0,2,4,3,6,2,7,7,9
%N Decimal expansion of Sum{k=2..infinity} (-1)^k/A165559(k).
%C Alternating sum of the reciprocals of the partial products of the arithmetic derivatives.
%H Ray Chandler, <a href="/A209873/b209873.txt">Table of n, a(n) for n = 0..85</a> (corrected by Ray Chandler, Jan 19 2019)
%e 0.003472825...
%p with(numtheory);
%p P:=proc(i)
%p local a, b, f, n, p, pfs;
%p a:=0; b:=1;
%p for n from 2 by 1 to i do
%p f:= A003415(n);
%p b:=b*f; a:=a+(-1)^n/b;
%p od;
%p print(evalf(a, 300));
%p end:
%p P(1000);
%t digits = 84; d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; p[n_] := p[n] = Sum[(-1)^k/Product[d[j], {j, 2, k}], {k, 2, n}] // RealDigits[#, 10, digits] & // First; p[digits]; p[n = 2*digits]; While[p[n] != p[n/2], n = 2*n]; Join[{0, 0}, p[n]] (* _Jean-François Alcover_, Feb 21 2014 *)
%Y Cf. A003415, A190144, A190145, A190146, A190147.
%K nonn,cons
%O 0,3
%A _Paolo P. Lava_, Apr 02 2012
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