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A190146 Decimal expansion of Sum_{k>=2} (1/Sum_{j=2..k} j'), where n' is the arithmetic derivative of n.

%I A190146 #33 Sep 27 2023 13:19:49
%S A190146 2,3,3,0,0,9
%N A190146 Decimal expansion of Sum_{k>=2} (1/Sum_{j=2..k} j'), where n' is the arithmetic derivative of n.
%C A190146 Slow convergence.
%C A190146 a(7) is likely either 3 or 4. Is there a simple proof that this sum converges? - _Nathaniel Johnston_, May 24 2011
%C A190146 From _Husnain Raza_, Aug 29 2023: (Start)
%C A190146 The series indeed converges: we have that the series is C = Sum_{k>=2} (1/Sum_{j=2..k} A003415(j)).
%C A190146 Let s_k = Sum_{j=2..k} A003415(j) be the inner sum.
%C A190146 It is known that s_k = (1/2)*T_0*k^2 + O(k^(1+n)) for all real n > 0 where T_0 = A136141.
%C A190146 Therefore, 1/s_k = (2/T_0)*k^(-2) + O(k^(-3+n)) = (2/T_0)*k^(-2) + O(k^(-3)).
%C A190146 Summing both sides from k=2 to infinity, we have that:
%C A190146 C = Sum_{k >= 2} 1/s_k = Sum_{k >= 2} ((2/T_0)*k^(-2) + O(k^(-3))), which converges. (End)
%e A190146 1/2' + 1/(2'+3') + 1/(2'+3'+4') + 1/(2'+3'+4'+5') + ... = 1 + 1/2 + 1/6 + 1/7 + ... = 2.33009...
%p A190146 with(numtheory);
%p A190146 P:=proc(i)
%p A190146 local a,b,f,n,p,pfs;
%p A190146 a:=0; b:=0;
%p A190146 for n from 2 to i do
%p A190146   pfs:=ifactors(n)[2];
%p A190146   f:=n*add(op(2,p)/op(1,p),p=pfs);
%p A190146   b:=b+f; a:=a+1/b;
%p A190146 od;
%p A190146 print(evalf(a,300));
%p A190146 end:
%p A190146 P(1000);
%t A190146 digits = 5; d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; p[m_] := p[m] = Sum[1/Sum[d[j], {j, 2, k}], {k, 2, m}] // RealDigits[#, 10, digits]& // First; p[digits]; p[m = 2*digits]; While[Print["p(", m, ") = ", p[m]]; p[m] != p[m/2], m = 2*m]; p[m] (* _Jean-François Alcover_, Feb 21 2014 *)
%Y A190146 Cf. A003415, A190144, A190145, A190147.
%K A190146 nonn,more,cons
%O A190146 1,1
%A A190146 _Paolo P. Lava_, May 05 2011
%E A190146 a(6) corrected and a(7) removed by _Nathaniel Johnston_, May 24 2011