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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).

%I A304661 #33 Sep 04 2018 20:56:37
%S A304661 1,8,9,12,16,18,24,27,32,36,40,45,48,50,54,60,64,72,75,80,81,90,96,
%T A304661 100,108,120,125,128,135,140,144,147,150,160,162,168,175,180,189,192,
%U A304661 196,200,210,216,224,225,240,243,245,250,252,256,270,280,288,294,300
%N 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).
%C A304661 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 A304661 These numbers are analogous to numbers that are "sqrt(n-1)-smooth" (see A063539).
%H A304661 Alois P. Heinz, <a href="/A304661/b304661.txt">Table of n, a(n) for n = 1..1000</a>
%e A304661 40 = 2^3*5 is a term because 2 and 5 are both less than log_2(40).
%e A304661 63 = 9*7 is not a term since 7 is greater than log_2(63).
%e A304661 1 is vacuously a term since it has no prime factors.
%p A304661 a:= proc(n) option remember; local k; for k from 1+a(n-1) while {}<>
%p A304661       select(x-> is(x>=log[2](k)), numtheory[factorset](k)) do od; k
%p A304661     end: a(1):=1:
%p A304661 seq(a(n), n=1..100);  # _Alois P. Heinz_, May 18 2018
%t A304661 Join[{1},Select[Range[300],FactorInteger[#][[-1,1]]<Log2[#]&]] (* _Harvey P. Dale_, Sep 04 2018 *)
%o A304661 (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 A304661 Cf. A063539.
%K A304661 nonn
%O A304661 1,2
%A A304661 _Richard Locke Peterson_, May 16 2018