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A273487 Density of numbers without prime exponents in their factorization.

%I A273487 #22 Jul 11 2016 11:49:13
%S A273487 6,5,0,4,4,5,6,0,8,4,2,1,9,1,2,6,9,1,3,9,0,4,4,4,3,6,1,1,0,4,6,5,9,6,
%T A273487 4,5,5,7,7,0,1,0,2,9,6,9,2,2,0,5,4,9,7,6,0,2,0,1,9,3,5,8,8,5,5,5,2,3,
%U A273487 4,2,8,6,9,1,6,8,2,1,3,6,7,7,4,9,3
%N A273487 Density of numbers without prime exponents in their factorization.
%F A273487 Prod_{p prime} 1 - (1 - 1/p)*Sum_{q prime} p^-q.
%e A273487 0.6504456084219126913904443611046...
%p A273487 eser := 1-x^2+x^4 ;
%p A273487 for pidx from 3 to 100 do
%p A273487     p := ithprime(pidx) ;
%p A273487     eser := eser -x^p+x^(p+1) ;
%p A273487 end do:
%p A273487 eser := taylor(eser,x=0,p) ;
%p A273487 gfun[seriestolist](eser) ;
%p A273487 subsop(1=NULL,%) ;
%p A273487 L := EULERi(%) ;
%p A273487 Digits := 180 ;
%p A273487 x := 1.0 ;
%p A273487 for i from 2 to nops(L) do
%p A273487     if op(i,L) <> 0 then
%p A273487         x := x*evalf(Zeta(i)^op(i,L)) ;
%p A273487         printf("%.70f\n",x) ;
%p A273487     fi ;
%p A273487 end do; # _R. J. Mathar_, Jul 11 2016
%o A273487 (PARI) leps=log(2)*(1-bitprecision(1.))
%o A273487 f(x)=my(s=0.);forprime(p=2,1-leps/log(x),s+=x^-p);s
%o A273487 6/Pi^2*prodeuler(p=2,1e6,(1-(1-1/p)*f(p))/(1-1/p^2))
%Y A273487 Density of A274034.
%K A273487 nonn,cons
%O A273487 0,1
%A A273487 _Charles R Greathouse IV_, Jul 01 2016