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

2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 101, 102

Offset: 1

Author

Dean Hickerson, Aug 28 2001

Keywords

• nonn
• easy

Comments

This set (S say) has density d(S) = Log(2) - Benoit Cloitre, Jun 12 2002

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

Examples

			9=3*3 is not "jagged", but 10=5*2 is "jagged": 5 > sqrt(10).
20=5*2*2 is "jagged", but not squarefree, cf. A005117.
		

References

• Steven R. Finch, Mathematical Constants, 2003, chapter 2.21.

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
• Eric Weisstein's World of Mathematics, Greatest Prime Factor

Crossrefs

Cf. A048098, A063538, A063539.

Programs

Mathematica

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 *)

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

Python

from math import isqrt
from sympy import primepi
def A064052(n):
    def f(x): return int(n+x-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