Distribution of Primes

Primes

\(\(\(Prime[5]\)\(
\)
    \)\)

\(\(\(Table[Prime[n], {n, 100}]\)\(
\)
    \)\)

\(\(\(FactorInteger[14151353]\)\(
\)
    \)\)

\(\(\(PrimePi[1000000]\)\(
\)
    \)\)

f[x_] :=  x/Log[x]
\(Table[
\ \ {10^k, PrimePi[10^k], N[f[10^k]], 
          N[PrimePi[10^k]/f[10^k]]}, 
\ \ {k, 1, 6}
] // 
      TableForm\)

Do[n = Prime[k]; np = Prime[k + 1]; npp = Prime[k + 2]; 
      If[npp - np == np - n, Print[{n, np, npp, np - n}]], {k, 1, 
        200}]