A304661 - OEIS

%I #33 Sep 04 2018 20:56:37

%S 1,8,9,12,16,18,24,27,32,36,40,45,48,50,54,60,64,72,75,80,81,90,96,

%T 100,108,120,125,128,135,140,144,147,150,160,162,168,175,180,189,192,

%U 196,200,210,216,224,225,240,243,245,250,252,256,270,280,288,294,300

%N 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).

%C 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.

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

%H Alois P. Heinz, <a href="/A304661/b304661.txt">Table of n, a(n) for n = 1..1000</a>

%e 40 = 2^3*5 is a term because 2 and 5 are both less than log_2(40).

%e 63 = 9*7 is not a term since 7 is greater than log_2(63).

%e 1 is vacuously a term since it has no prime factors.

%p a:= proc(n) option remember; local k; for k from 1+a(n-1) while {}<>

%p       select(x-> is(x>=log[2](k)), numtheory[factorset](k)) do od; k

%p     end: a(1):=1:

%p seq(a(n), n=1..100);  # _Alois P. Heinz_, May 18 2018

%t Join[{1},Select[Range[300],FactorInteger[#][[-1,1]]<Log2[#]&]] (* _Harvey P. Dale_, Sep 04 2018 *)

%o (PARI) isok(n) = my(f=factor(n)[,1], z = log(n)/log(2)); #select(x->(x >= z), f) == 0; \\ _Michel Marcus_, Jun 01 2018

%Y Cf. A063539.

%K nonn

%O 1,2

%A _Richard Locke Peterson_, May 16 2018