cp's OEIS Frontend

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

%I A137845 #29 May 11 2020 02:05:45
%S A137845 8,16,24,27,32,36,48,54,64,72,81,96,108,128,144,150,160,162,180,192,
%T A137845 200,216,225,240,243,250,256,270,288,300,320,324,360,375,384,400,405,
%U A137845 432,450,480,486,500,512,540,576,600,625,640,648,675,720,729,750,768,800
%N A137845 Logarithmically smooth numbers; numbers n whose largest prime factor is less than log(n).
%C A137845 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 A137845 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 A137845 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 A137845 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 A137845 48 = 2^4 * 3, and log(48) = 3.8712... > 3. Hence 48 is in the sequence.
%e A137845 49 = 7^2 but log(49) = 3.89182... < 7, so 49 is not in the sequence.
%t A137845 Select[Range[2,1000], FactorInteger[#][[-1,1]] < Log[#] &]
%o A137845 (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 A137845 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 A137845 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 A137845 Cf. A048098, A063539 (two versions of Sqrt-smooth numbers).
%Y A137845 See also A333534.
%K A137845 nonn
%O A137845 1,1
%A A137845 _T. D. Noe_, Feb 14 2008