A096300 Number of positive integers <= n with no prime factor > log(n).

1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18

Offset: 1

Author

Ralf Stephan, Aug 03 2004

Keywords

• nonn
• easy

Links

• A. Granville, On positive integers <= x with prime factors <= t log x, Number Theory and Applications (ed. R.A Mollin), (Kluwer, NATO ASI, 1989), 403-422.

Programs

Mathematica

a[n_] := Select[Range[n], FactorInteger[#][[-1, 1]] <= Log[n]&] // Length;
a[1] = a[2] = 1;
a /@ Range[75] (* Jean-François Alcover, May 17 2020 *)

PARI

a(n)=local(s, t); s=0; for(k=1, n, f=factor(k); t=0; for(l=1, matsize(f)[1], if(f[l, 1]>log(n), t=1; break)); s=s+!t); s

Formula

From Charlie Neder, Feb 08 2019: (Start)

a(n) = A000012(n) for 0 < n <= floor(e^2) = 7,

A070939(n) for 7 < n <= floor(e^3) = 20,

A071521(n) for 20 < n <= floor(e^5) = 148,

A071520(n) for 148 < n <= floor(e^7) = 1096,

A071604(n) for 1096 < n <= floor(e^11) = 59874,

and so on. (End)