A083269 a(n) = pi(A052180(n)) = A000720(A052180(n)); subscript of last prime used in Eratosthenes sieve by which all composites between n-th and (n+1)th primes were excluded.

0, 1, 1, 2, 1, 2, 1, 2, 3, 1, 3, 2, 1, 2, 4, 3, 1, 3, 2, 1, 4, 2, 3, 4, 2, 1, 2, 1, 2, 5, 2, 4, 1, 5, 1, 3, 4, 2, 6, 3, 1, 5, 1, 2, 1, 5, 6, 2, 1, 2, 3, 1, 6, 5, 4, 3, 1, 3, 2, 1, 7, 6, 2, 1, 2, 7, 3, 5, 1, 2, 3, 8, 4, 6, 2, 3, 7, 2, 6, 4, 1, 4, 1, 8, 2, 3, 5, 2, 1, 2, 5, 6, 2, 7, 2, 3, 5, 1, 9, 3, 8, 6, 3, 1, 3

Offset: 1

Author

Labos Elemer, May 14 2003

Keywords

• nonn

Examples

			Of composites between the 24th and 25th primes (89, 97), the least prime divisors are {2,7,2,3,2,5,2}.
The largest of these is 7. This means that pi(7)=4 steps in prime sieving are required to sweep out all composites between 89 and 97: {90,92,94,96}, {93}, {95}, and {91} were excluded in the 1st, 2nd, 3rd, and 4th steps, respectively.
So a(24)=4.
		

Crossrefs

Cf. A052180, A000720.

Programs

Mathematica

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] mi[x_] := Min[ba[x]] Table[PrimePi[Max[Table[mi[ba[w]], {w, Prime[j]+1, -1+Prime[j+1]}]]], {j, 1, 30}]