cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

T. D. Noe, Feb 14 2008

Keywords

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

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

A333534 a(n) is the number of log(n)-smooth numbers <= n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19
Offset: 2

Views

Author

N. J. A. Sloane, Apr 08 2020

Keywords

Comments

Number of k <= n such that the greatest prime factor of k is <= log(n).

Links

Crossrefs

Cf. A001671, A006530, A051102, A137845, A295084.

Programs

  • Maple
    A333534 := n -> nops(select(k -> A006530(k) <= ilog(n), [$1..n])):
seq(A333534(n), n=2..86); # Peter Luschny, Apr 09 2020
# second Maple program:
b:= proc(n) option remember; max(1, map(i-> i[1], ifactors(n)[2])) end:
a:= n-> (t-> add(`if`(b(i)<= t, 1, 0), i=1..n))(ilog(n)):
seq(a(n), n=2..100);  # Alois P. Heinz, Apr 09 2020
  • Mathematica
    a[n_] := Select[Range[n], FactorInteger[#][[-1, 1]] <= Log[n]&] // Length;
a /@ Range[2, 100] (* Jean-François Alcover, May 17 2020 *)
  • PARI
    gpf(j)={if(j==1,1,my(f=factor(j));f[#f[,2],1])};
for(n=2,80,my(L=log(n));print1(sum(k=1,n,gpf(k)<=L),", ")) \\ Hugo Pfoertner, Apr 09 2020
  • PARI
    sm(lim, p)=if(p==2, return(logint(lim\1, 2)+1)); my(s=0, q=precprime(p-1), t=1); for(e=0, logint(lim\=1, p), s+=sm(lim\t, q); t*=p); s
a(n)=if(n<8,return(n>2)); sm(n, precprime(log(n))) \\ Charles R Greathouse IV, Apr 16 2020

Formula

a(n) = A096300(n), n>2. - R. J. Mathar, Apr 27 2020
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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