A273487 Density of numbers without prime exponents in their factorization.

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

Offset: 0

Author

Charles R Greathouse IV, Jul 01 2016

Keywords

• nonn
• cons

Examples

			0.6504456084219126913904443611046...
		

Crossrefs

Density of A274034.

Programs

Maple

eser := 1-x^2+x^4 ;
for pidx from 3 to 100 do
    p := ithprime(pidx) ;
    eser := eser -x^p+x^(p+1) ;
end do:
eser := taylor(eser,x=0,p) ;
gfun[seriestolist](eser) ;
subsop(1=NULL,%) ;
L := EULERi(%) ;
Digits := 180 ;
x := 1.0 ;
for i from 2 to nops(L) do
    if op(i,L) <> 0 then
        x := x*evalf(Zeta(i)^op(i,L)) ;
        printf("%.70f\n",x) ;
    fi ;
end do; # R. J. Mathar, Jul 11 2016

PARI

leps=log(2)*(1-bitprecision(1.))
f(x)=my(s=0.);forprime(p=2,1-leps/log(x),s+=x^-p);s
6/Pi^2*prodeuler(p=2,1e6,(1-(1-1/p)*f(p))/(1-1/p^2))

Formula

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