A304661 Numbers n that are log_2(n-1)-smooth, i.e., such that all the prime factors of n are less than log_2(n).

1, 8, 9, 12, 16, 18, 24, 27, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 81, 90, 96, 100, 108, 120, 125, 128, 135, 140, 144, 147, 150, 160, 162, 168, 175, 180, 189, 192, 196, 200, 210, 216, 224, 225, 240, 243, 245, 250, 252, 256, 270, 280, 288, 294, 300

Offset: 1

Author

Richard Locke Peterson, May 16 2018

Keywords

• nonn

Comments

The sequence is a monoid since it contains 1 and is closed under multiplication, since if m and n are terms, then any prime dividing m or n must be less than log base 2 of m*n. Density: 27% of the numbers from 1 to 64 are terms. From 2^120 +1 to 2^120+64, 0% are terms. However, it is an infinite sequence, since 2^n is always a term, for n>2.

These numbers are analogous to numbers that are "sqrt(n-1)-smooth" (see A063539).

Examples

			40 = 2^3*5 is a term because 2 and 5 are both less than log_2(40).
63 = 9*7 is not a term since 7 is greater than log_2(63).
1 is vacuously a term since it has no prime factors.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A063539.

Programs

Maple

a:= proc(n) option remember; local k; for k from 1+a(n-1) while {}<>
      select(x-> is(x>=log[2](k)), numtheory[factorset](k)) do od; k
    end: a(1):=1:
seq(a(n), n=1..100);  # Alois P. Heinz, May 18 2018

Mathematica

Join[{1},Select[Range[300],FactorInteger[#][[-1,1]]Harvey P. Dale, Sep 04 2018 *)

PARI

isok(n) = my(f=factor(n)[,1], z = log(n)/log(2)); #select(x->(x >= z), f) == 0; \\ Michel Marcus, Jun 01 2018