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Density of numbers without prime exponents in their factorization.
1

%I #22 Jul 11 2016 11:49:13

%S 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 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 4,2,8,6,9,1,6,8,2,1,3,6,7,7,4,9,3

%N Density of numbers without prime exponents in their factorization.

%F Prod_{p prime} 1 - (1 - 1/p)*Sum_{q prime} p^-q.

%e 0.6504456084219126913904443611046...

%p eser := 1-x^2+x^4 ;

%p for pidx from 3 to 100 do

%p p := ithprime(pidx) ;

%p eser := eser -x^p+x^(p+1) ;

%p end do:

%p eser := taylor(eser,x=0,p) ;

%p gfun[seriestolist](eser) ;

%p subsop(1=NULL,%) ;

%p L := EULERi(%) ;

%p Digits := 180 ;

%p x := 1.0 ;

%p for i from 2 to nops(L) do

%p if op(i,L) <> 0 then

%p x := x*evalf(Zeta(i)^op(i,L)) ;

%p printf("%.70f\n",x) ;

%p fi ;

%p end do; # _R. J. Mathar_, Jul 11 2016

%o (PARI) leps=log(2)*(1-bitprecision(1.))

%o f(x)=my(s=0.);forprime(p=2,1-leps/log(x),s+=x^-p);s

%o 6/Pi^2*prodeuler(p=2,1e6,(1-(1-1/p)*f(p))/(1-1/p^2))

%Y Density of A274034.

%K nonn,cons

%O 0,1

%A _Charles R Greathouse IV_, Jul 01 2016