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

1,2

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

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 *)

Prog

(PARI)
print1(1, ", "); for(n=2, 1000, if(vecmax(factor(n)[, 1])<=sqrt(n), print1(n, ", ")))
(Haskell)
a048098 n = a048098_list !! (n-1)
a048098_list = [x | x <- [1..], a006530 x ^ 2 <= x]
-- Reinhard Zumkeller, Oct 12 2011
(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

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.

Sequence in context: A278517 A053443 A360070 * A322109 A122145 A328014

Adjacent sequences: A048095 A048096 A048097 * A048099 A048100 A048101

Keyword

easy,nonn,nice

Author

J. Lowell

Extensions

More terms from James A. Sellers, Apr 22 2000

Edited by Charles R Greathouse IV, Nov 08 2010

Status

approved