|
| |
|
|
A137845
|
|
Logarithmically smooth numbers; numbers n whose largest prime factor is less than log(n).
|
|
5
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
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
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
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.
|
|
|
MATHEMATICA
|
Select[Range[2, 1000], FactorInteger[#][[-1, 1]] < Log[#] &]
|
|
|
PROG
|
(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
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
