A190146 - OEIS

%I #33 Sep 27 2023 13:19:49

%S 2,3,3,0,0,9

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

%C Slow convergence.

%C a(7) is likely either 3 or 4. Is there a simple proof that this sum converges? - _Nathaniel Johnston_, May 24 2011

%C From _Husnain Raza_, Aug 29 2023: (Start)

%C The series indeed converges: we have that the series is C = Sum_{k>=2} (1/Sum_{j=2..k} A003415(j)).

%C Let s_k = Sum_{j=2..k} A003415(j) be the inner sum.

%C 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 Therefore, 1/s_k = (2/T_0)*k^(-2) + O(k^(-3+n)) = (2/T_0)*k^(-2) + O(k^(-3)).

%C Summing both sides from k=2 to infinity, we have that:

%C C = Sum_{k >= 2} 1/s_k = Sum_{k >= 2} ((2/T_0)*k^(-2) + O(k^(-3))), which converges. (End)

%e 1/2' + 1/(2'+3') + 1/(2'+3'+4') + 1/(2'+3'+4'+5') + ... = 1 + 1/2 + 1/6 + 1/7 + ... = 2.33009...

%p with(numtheory);

%p P:=proc(i)

%p local a,b,f,n,p,pfs;

%p a:=0; b:=0;

%p for n from 2 to i do

%p   pfs:=ifactors(n)[2];

%p   f:=n*add(op(2,p)/op(1,p),p=pfs);

%p   b:=b+f; a:=a+1/b;

%p od;

%p print(evalf(a,300));

%p end:

%p P(1000);

%t 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 Cf. A003415, A190144, A190145, A190147.

%K nonn,more,cons

%O 1,1

%A _Paolo P. Lava_, May 05 2011

%E a(6) corrected and a(7) removed by _Nathaniel Johnston_, May 24 2011