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%I #11 Feb 21 2014 04:55:34
%S 1,5,0,7,8,1,0,6,6,7,6,2,2,8,9,8,2,8,3,8,3,3,1,5,3,9,0,3,7,6,5,3,7,7,
%T 7,2,7,2,4,7,3,4,6,8,8,5,1,9,3,8,9,5,5,8,5,5,3,1,9,1,3,9,0,8,6,3,0,1,
%U 2,5,3,8,1,3,3,9,5,8,9,8,9,1,1,6,7,1,4,7,5,0,5,2,5,1,0,6,3,0,6,1,3,1,7,1,2,7,1,9,4,9,9,2,2,7,3,6,6,2,4,9
%N Decimal expansion of Sum{k=1..infinity}(1/Sum{j=1..k} j^j’), where n’ is the arithmetic derivative of n.
%e 1/1^1’+1/(1^1’+2^2’)+1/(1^1’+2^2’+3^3’)+1/(1^1’+2^2’+3^3’+4^4’)+... = 1+1/3+1/6+1/262+... = 1.50781066762289...
%p with(numtheory);
%p P:=proc(i)
%p local a,b,f,n,p,pfs;
%p a:=0; b:=0;
%p for n from 1 by 1 to i do
%p pfs:=ifactors(n)[2];
%p f:=n*add(op(2,p)/op(1,p),p=pfs);
%p b:=b+n^f; a:=a+1/b;
%p od;
%p print(evalf(a,300));
%p end:
%p P(1000);
%t digits = 120; d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; p[m_] := p[m] = Sum[1/Sum[j^d[j], {j, 1, k}], {k, 1, m}] // RealDigits[#, 10, digits]& // First; p[digits]; p[m = 2*digits]; While[p[m] != p[m/2], m = 2*m]; p[m] (* _Jean-François Alcover_, Feb 21 2014 *)
%Y Cf. A003415, A190144, A190145, A190146.
%K nonn,cons
%O 1,2
%A _Paolo P. Lava_, May 05 2011
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