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A190144 Decimal expansion of Sum_{k>=2} (1/Product_{j=2..k} j'), where n' is the arithmetic derivative of n.

%I A190144 #26 May 26 2025 00:25:49
%S A190144 2,6,0,5,0,7,2,7,0,5,2,9,7,3,2,2,8,7,0,8,0,3,4,2,6,4,1,2,4,1,8,3,8,7,
%T A190144 8,5,1,3,7,0,8,5,7,3,6,3,2,7,6,6,3,7,2,2,4,3,8,5,8,5,0,8,4,0,7,3,1,0,
%U A190144 5,7,5,9,3,7,1,6,1,9,7,5,1,7,0,4,7,7,4,9,9,4,5,4,7,4,8,4,5,6,1,7,0,8,8,9,4,7,7,6,2,0,9,5,9,7,8,5,2,4,4,7
%N A190144 Decimal expansion of Sum_{k>=2} (1/Product_{j=2..k} j'), where n' is the arithmetic derivative of n.
%C A190144 This constant differs by 0.11320912... from the formally similar expansion of e, Sum_{n>=0} 1/n!.
%e A190144 1/2' + 1/(2' * 3') + 1/(2' * 3' * 4') + 1/(2' * 3' * 4' * 5') + 1/(2' * 3' * 4' * 5' * 6') + ... = 1 + 1 + 1/4 + 1/4 + 1/20 + ... = 2.605072705297...
%p A190144 with(numtheory);
%p A190144 P:=proc(i)
%p A190144 local a,b,f,n,p,pfs;
%p A190144 a:=0; b:=1;
%p A190144 for n from 2 by 1 to i do
%p A190144   pfs:=ifactors(n)[2];
%p A190144   f:=n*add(op(2,p)/op(1,p),p=pfs);
%p A190144   b:=b*f; a:=a+1/b;
%p A190144 od;
%p A190144 print(evalf(a,300));
%p A190144 end:
%p A190144 P(1000);
%t A190144 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[d[j], {j, 2, k}], {k, 2, 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 A190144 Cf. A003415, A190145, A190146, A190147.
%K A190144 nonn,cons
%O A190144 1,1
%A A190144 _Paolo P. Lava_, May 05 2011