A063762 Sqrt(n)-rough nonprimes: largest prime factor of n (A006530) >= sqrt(n).

4, 6, 9, 10, 14, 15, 20, 21, 22, 25, 26, 28, 33, 34, 35, 38, 39, 42, 44, 46, 49, 51, 52, 55, 57, 58, 62, 65, 66, 68, 69, 74, 76, 77, 78, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 99, 102, 104, 106, 110, 111, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 129, 130, 133

Offset: 1

Author

Robert G. Wilson v, Aug 14 2001

Keywords

• nonn

Comments

A positive integer is called y-rough if all its prime factors are >= y.

References

• D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms; see pp. 95-98.

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
• Beeler, M., Gosper, R. W. and Schroeppel, R., HAKMEM, ITEM 29

Crossrefs

Cf. A006530, A063538, A063539, A063763.

Programs

Mathematica

Select[ Range[ 2, 150 ], !PrimeQ[ # ] && FactorInteger[ # ] [ [ -1, 1 ] ] >= Sqrt[ # ] & ]

PARI

{ n=0; for (m=2, 10^9, f=vecmax(component(factor(m), 1)); if(!isprime(m) && f^2 >= m, write("b063762.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 30 2009

Python

from math import isqrt
from sympy import primepi
def A063762(n):
    def f(x): return int(n+(x if x<=3 else x-primepi(x//(y:=isqrt(x)))-sum(primepi(x//i)-primepi(i) for i in range(2,y))))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 05 2024