A316350 Positive integers x that are x/log(x) smooth, that is, if a prime p divides x, then p <= x/log(x).

1, 2, 4, 6, 8, 9, 12, 15, 16, 18, 20, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 44, 45, 48, 49, 50, 52, 54, 55, 56, 60, 63, 64, 65, 66, 70, 72, 75, 77, 78, 80, 81, 84, 85, 88, 90, 91, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 115, 117, 119, 120

Offset: 1

Author

Richard Locke Peterson, Jun 29 2018

Keywords

• nonn

Comments

This sequence is a monoid under multiplication, since if x and y are terms in the sequence and p < x/log(x), then p < xy/log(xy). However, if a term in the sequence is multiplied by a number outside the sequence, the result need not be in the sequence.

Examples

			1 is in the sequence because no primes divide 1, 2 is in the sequence since 2 divides 2 and 2 < 2/log(2) ~ 2.9, but 10 is not in the sequence since 5 divides 10 and 5 is not less than 10/log(10) ~ 4.34.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A050500.

Programs

Maple

filter:= n -> is(max(numtheory:-factorset(n))Robert Israel, Oct 21 2021

Mathematica

ok[n_] := AllTrue[First /@ FactorInteger[n], # Log[n] <= n &]; Select[ Range[120], ok] (* Giovanni Resta, Jun 30 2018 *)

PARI

isok(n) = my(f=factor(n)); for (k=1, #f~, if (f[k,1] >= n/log(n), return(0))); return (1); \\ Michel Marcus, Jul 02 2018