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A064052 Not sqrt(n)-smooth: some prime factor of n is > sqrt(n).

%I A064052 #42 Feb 16 2025 08:32:45
%S A064052 2,3,5,6,7,10,11,13,14,15,17,19,20,21,22,23,26,28,29,31,33,34,35,37,
%T A064052 38,39,41,42,43,44,46,47,51,52,53,55,57,58,59,61,62,65,66,67,68,69,71,
%U A064052 73,74,76,77,78,79,82,83,85,86,87,88,89,91,92,93,94,95,97,99,101,102
%N A064052 Not sqrt(n)-smooth: some prime factor of n is > sqrt(n).
%C A064052 This set (S say) has density d(S) = Log(2) - _Benoit Cloitre_, Jun 12 2002
%C A064052 Finch defines a positive integer N to be "jagged" if its largest prime factor is > sqrt(N). - _Frank Ellermann_, Apr 21 2011
%D A064052 Steven R. Finch, Mathematical Constants, 2003, chapter 2.21.
%H A064052 Ray Chandler, <a href="/A064052/b064052.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Harry J. Smith)
%H A064052 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GreatestPrimeFactor.html">Greatest Prime Factor</a>
%e A064052 9=3*3 is not "jagged", but 10=5*2 is "jagged": 5 > sqrt(10).
%e A064052 20=5*2*2 is "jagged", but not squarefree, cf. A005117.
%t A064052 Reap[For[n = 2, n <= 102, n++, f = FactorInteger[n][[-1, 1]]; If[f > Sqrt[n], Sow[n]]]][[2, 1]] (* _Jean-François Alcover_, May 16 2014 *)
%o A064052 (PARI) { n=0; for (m=2, 10^9, f=factor(m)~; if (f[1, length(f)]^2 > m, write("b064052.txt", n++, " ", m); if (n==1000, break)) ) } \\ _Harry J. Smith_, Sep 06 2009
%o A064052 (Python)
%o A064052 from math import isqrt
%o A064052 from sympy import primepi
%o A064052 def A064052(n):
%o A064052     def f(x): return int(n+x-sum(primepi(x//i)-primepi(i) for i in range(1,isqrt(x)+1)))
%o A064052     def bisection(f,kmin=0,kmax=1):
%o A064052         while f(kmax) > kmax: kmax <<= 1
%o A064052         while kmax-kmin > 1:
%o A064052             kmid = kmax+kmin>>1
%o A064052             if f(kmid) <= kmid:
%o A064052                 kmax = kmid
%o A064052             else:
%o A064052                 kmin = kmid
%o A064052         return kmax
%o A064052     return bisection(f) # _Chai Wah Wu_, Sep 01 2024
%Y A064052 Cf. A048098, A063538, A063539.
%K A064052 nonn,easy
%O A064052 1,1
%A A064052 _Dean Hickerson_, Aug 28 2001