A137845 Logarithmically smooth numbers; numbers n whose largest prime factor is less than log(n).

8, 16, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 150, 160, 162, 180, 192, 200, 216, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 360, 375, 384, 400, 405, 432, 450, 480, 486, 500, 512, 540, 576, 600, 625, 640, 648, 675, 720, 729, 750, 768, 800

Offset: 1

Author

T. D. Noe, Feb 14 2008

Keywords

• nonn

Comments

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.

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

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

Examples

			48 = 2^4 * 3, and log(48) = 3.8712... > 3. Hence 48 is in the sequence.
49 = 7^2 but log(49) = 3.89182... < 7, so 49 is not in the sequence.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Crossrefs

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

See also A333534.

Programs

Mathematica

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

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)
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]
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