%I #37 Sep 01 2024 10:42:02

%S 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 38,39,41,42,43,44,46,47,51,52,53,55,57,58,59,61,62,65,66,67,68,69,71,

%U 73,74,76,77,78,79,82,83,85,86,87,88,89,91,92,93,94,95,97,99,101,102

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

%C This set (S say) has density d(S) = Log(2) - _Benoit Cloitre_, Jun 12 2002

%C Finch defines a positive integer N to be "jagged" if its largest prime factor is > sqrt(N). - _Frank Ellermann_, Apr 21 2011

%D S. R. Finch, Mathematical Constants, 2003, chapter 2.21.

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GreatestPrimeFactor.html">Greatest Prime Factor</a>

%e 9=3*3 is not "jagged", but 10=5*2 is "jagged": 5 > sqrt(10).

%e 20=5*2*2 is "jagged", but not squarefree, cf. A005117.

%t 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 (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 (Python)

%o from math import isqrt

%o from sympy import primepi

%o def A064052(n):

%o def f(x): return int(n+x-sum(primepi(x//i)-primepi(i) for i in range(1,isqrt(x)+1)))

%o def bisection(f,kmin=0,kmax=1):

%o while f(kmax) > kmax: kmax <<= 1

%o while kmax-kmin > 1:

%o kmid = kmax+kmin>>1

%o if f(kmid) <= kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o return kmax

%o return bisection(f) # _Chai Wah Wu_, Sep 01 2024

%Y Cf. A048098, A063538, A063539.

%K nonn,easy

%O 1,1

%A _Dean Hickerson_, Aug 28 2001