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A190145 Decimal expansion of Sum_{k>=1} (1/Product_{j=1..k} j^j'), where n' is the arithmetic derivative of n.

%I A190145 #13 Jan 03 2021 12:30:00
%S A190145 1,6,6,7,4,4,7,9,3,5,8,0,3,6,9,3,9,0,5,5,8,0,5,6,7,5,1,2,8,7,3,3,9,6,
%T A190145 0,2,0,3,9,4,4,7,8,0,1,1,3,8,1,7,7,1,6,7,0,0,5,3,7,2,8,2,2,7,6,2,0,5,
%U A190145 8,9,3,2,9,0,2,5,2,7,7,9,1,7,0,2,4,5,2,5,4,9,9,7,7,0,8,8,1,2,2,4,8,2,4,1,6,2,6,3,3,6,8,6,1,5,1,1,1,8,9,1
%N A190145 Decimal expansion of Sum_{k>=1} (1/Product_{j=1..k} j^j'), where n' is the arithmetic derivative of n.
%D A190145 1
%e A190145 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 + 1/2 + 1/6 + 1/1536 + ... = 1.6674479358036...
%p A190145 with(numtheory);
%p A190145 P:=proc(i)
%p A190145 local a,b,c,d,f,n,p,pfs,s;
%p A190145 a:=0; b:=1;
%p A190145 for n from 2 by 1 to i do
%p A190145   pfs:=ifactors(n)[2];
%p A190145   f:=n*add(op(2,p)/op(1,p),p=pfs);
%p A190145   b:=b*n^f; a:=a+1/b;
%p A190145 od;
%p A190145 print(evalf(a,300));
%p A190145 end:
%p A190145 P(1000);
%t A190145 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/Product[j^d[j], {j, 2, 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 A190145 Cf. A003415, A190144, A190146, A190147.
%K A190145 nonn,cons
%O A190145 1,2
%A A190145 _Paolo P. Lava_, May 05 2011