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

%I A048098 #60 Jul 02 2025 16:01:57
%S A048098 1,4,8,9,12,16,18,24,25,27,30,32,36,40,45,48,49,50,54,56,60,63,64,70,
%T A048098 72,75,80,81,84,90,96,98,100,105,108,112,120,121,125,126,128,132,135,
%U A048098 140,144,147,150,154,160,162,165,168,169,175,176,180,182,189,192,195
%N A048098 Numbers k that are sqrt(k)-smooth: if p | k then p^2 <= k when p is prime.
%C A048098 A006530(a(n))^2 <= a(n). - _Reinhard Zumkeller_, Oct 12 2011
%C A048098 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
%H A048098 T. D. Noe and William A. Tedeschi, <a href="/A048098/b048098.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms computed by T. D. Noe)
%H A048098 H. Davenport and P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1951-07.pdf">On sequences of positive integers</a>, J. Indian Math. Soc. 15 (1951), pp. 19-24.
%H A048098 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GreatestPrimeFactor.html">Greatest Prime Factor</a>
%H A048098 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RoundNumber.html">Round Number</a>
%t A048098 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 *)
%o A048098 (PARI)
%o A048098 print1(1, ", ");for(n=2, 1000, if(vecmax(factor(n)[, 1])<=sqrt(n), print1(n, ", ")))
%o A048098 (Haskell)
%o A048098 a048098 n = a048098_list !! (n-1)
%o A048098 a048098_list = [x | x <- [1..], a006530 x ^ 2 <= x]
%o A048098 -- _Reinhard Zumkeller_, Oct 12 2011
%o A048098 (Python)
%o A048098 from sympy import factorint
%o A048098 def ok(n):
%o A048098     return n == 1 if n < 2 else max(factorint(n))**2 <= n
%o A048098 print([k for k in range(196) if ok(k)]) # _Michael S. Branicky_, Dec 22 2021
%o A048098 (Python)
%o A048098 from math import isqrt
%o A048098 from sympy import primepi
%o A048098 def A048098(n):
%o A048098     def f(x): return int(n+sum(primepi(x//i)-primepi(i) for i in range(1,isqrt(x)+1)))
%o A048098     def bisection(f,kmin=0,kmax=1):
%o A048098         while f(kmax) > kmax: kmax <<= 1
%o A048098         while kmax-kmin > 1:
%o A048098             kmid = kmax+kmin>>1
%o A048098             if f(kmid) <= kmid:
%o A048098                 kmax = kmid
%o A048098             else:
%o A048098                 kmin = kmid
%o A048098         return kmax
%o A048098     return bisection(f) # _Chai Wah Wu_, Sep 01 2024
%Y A048098 Set union of A063539 and A001248.
%Y A048098 Cf. A006530, A063538, A063762, A063763, A064052.
%Y A048098 The following are all different versions of sqrt(n)-smooth numbers: A048098, A063539, A064775, A295084, A333535, A333536.
%K A048098 easy,nonn,nice
%O A048098 1,2
%A A048098 _J. Lowell_
%E A048098 More terms from _James Sellers_, Apr 22 2000
%E A048098 Edited by _Charles R Greathouse IV_, Nov 08 2010