A048098 Numbers k that are sqrt(k)-smooth: if p | k then p^2 <= k when p is prime.

1, 4, 8, 9, 12, 16, 18, 24, 25, 27, 30, 32, 36, 40, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 121, 125, 126, 128, 132, 135, 140, 144, 147, 150, 154, 160, 162, 165, 168, 169, 175, 176, 180, 182, 189, 192, 195

Offset: 1

Author

J. Lowell

Keywords

• easy
• nonn
• nice

Comments

A006530(a(n))^2 <= a(n). - Reinhard Zumkeller, Oct 12 2011

This set (say S) has density d(S) = 1-log(2) and multiplicative density m(S) = 1-exp(-Gamma). Multiplicative density: let A be a set of numbers, A(x) = { k in A | gpf(k) <=x } where gpf(k) denotes the greatest prime factor of k and let m(x)(A) = Product_{p<=x} (1 - 1/p)*Sum_{k in A(x)} 1/k. If lim_{x->infinity} m(x)(A) exists = m(A), this limit is called "multiplicative density" of A (Erdős and Davenport, 1951). - Benoit Cloitre, Jun 12 2002

Links

• T. D. Noe and William A. Tedeschi, Table of n, a(n) for n = 1..10000 (first 1000 terms computed by T. D. Noe)
• H. Davenport and P. Erdős, On sequences of positive integers, J. Indian Math. Soc. 15 (1951), pp. 19-24.
• Eric Weisstein's World of Mathematics, Greatest Prime Factor
• Eric Weisstein's World of Mathematics, Round Number

Crossrefs

Set union of A063539 and A001248.

Cf. A006530, A063538, A063762, A063763, A064052.

The following are all different versions of sqrt(n)-smooth numbers: A048098, A063539, A064775, A295084, A333535, A333536.

Programs

Haskell

a048098 n = a048098_list !! (n-1)
a048098_list = [x | x <- [1..], a006530 x ^ 2 <= x]
-- Reinhard Zumkeller, Oct 12 2011

Mathematica

gpf[n_] := FactorInteger[n][[-1, 1]]; A048098 = {}; For[n = 1, n <= 200, n++, If[ gpf[n] <= Sqrt[n], AppendTo[ A048098, n] ] ]; A048098 (* Jean-François Alcover, Jan 26 2012 *)

PARI

print1(1, ", ");for(n=2, 1000, if(vecmax(factor(n)[, 1])<=sqrt(n), print1(n, ", ")))

Python

from sympy import factorint
def ok(n):
    return n == 1 if n < 2 else max(factorint(n))**2 <= n
print([k for k in range(196) if ok(k)]) # Michael S. Branicky, Dec 22 2021

Python

from math import isqrt
from sympy import primepi
def A048098(n):
    def f(x): return int(n+sum(primepi(x//i)-primepi(i) for i in range(1,isqrt(x)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f) # Chai Wah Wu, Sep 01 2024

Extensions

More terms from James Sellers, Apr 22 2000

Edited by Charles R Greathouse IV, Nov 08 2010