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

0,1

Links

Table of n, a(n) for n=0..84.

Formula

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

Example

0.6504456084219126913904443611046...

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

Prog

(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))

Crossrefs

Density of A274034.

Sequence in context: A010773 A099288 A256717 * A134103 A196621 A340443

Adjacent sequences: A273484 A273485 A273486 * A273488 A273489 A273490

Keyword

nonn,cons

Author

Charles R Greathouse IV, Jul 01 2016

Status

approved