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Logarithmically smooth numbers; numbers n whose largest prime factor is less than log(n).
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%I #29 May 11 2020 02:05:45

%S 8,16,24,27,32,36,48,54,64,72,81,96,108,128,144,150,160,162,180,192,

%T 200,216,225,240,243,250,256,270,288,300,320,324,360,375,384,400,405,

%U 432,450,480,486,500,512,540,576,600,625,640,648,675,720,729,750,768,800

%N Logarithmically smooth numbers; numbers n whose largest prime factor is less than log(n).

%C The graph of this sequence has inflections when n first exceeds exp(prime(k)) for some k. See A051102. It appears that (2400, 2401) and (4374, 4375) are the only consecutive numbers in this sequence. See A116486 for a slightly different definition of logarithmically smooth.

%C The sequence is closed under multiplication, since if x,y are sequence terms, and a prime p divides x, then p is less than log(x), which is less than log(xy). - _Richard Locke Peterson_, Apr 12 2020

%C The Euler phi function of a(n) need not be logarithmically smooth, since phi(27)=18. This differs from k-smooth numbers. - _Richard Locke Peterson_, May 09 2020

%H Alois P. Heinz, <a href="/A137845/b137845.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)

%e 48 = 2^4 * 3, and log(48) = 3.8712... > 3. Hence 48 is in the sequence.

%e 49 = 7^2 but log(49) = 3.89182... < 7, so 49 is not in the sequence.

%t Select[Range[2,1000], FactorInteger[#][[-1,1]] < Log[#] &]

%o (PARI) sm(N, p)=if(p==2, return(powers(2, logint(N, 2)))); my(v=[], q=precprime(p-1), t=1); for(e=0, logint(N, p), v=concat(v, sm(N\t, q)*t); t*=p); Set(v)

%o smCapped(N, p, lim)=my(v=sm(N\1,p), i); i=setsearch(v,lim\=1,1); if(i==0, i=setsearch(v,lim)+1); v[i..#v]

%o list(lim)=if(lim<8,return([])); my(P=primes([2,log(lim\=1)\1]),v=[]); for(i=2,#P, v=concat(v,smCapped(exp(P[i]),P[i-1],exp(P[i-1])))); v=concat(v,smCapped(lim,P[#P],exp(P[#P]))); v \\ _Charles R Greathouse IV_, Apr 16 2020

%Y Cf. A048098, A063539 (two versions of Sqrt-smooth numbers).

%Y See also A333534.

%K nonn

%O 1,1

%A _T. D. Noe_, Feb 14 2008