A111906 Numbers k such that fewer primes, among primes <= the largest prime dividing k, divide k than do not.

5, 7, 11, 13, 17, 19, 22, 23, 25, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 67, 68, 69, 71, 73, 74, 76, 77, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 101, 102, 103, 104, 106, 107, 109, 111, 113, 114, 115

Offset: 1

Author

Leroy Quet, Aug 19 2005

Keywords

• nonn

Examples

			22 is included because 11 is the largest prime dividing 22. And of the primes <= 11 (2,3,5,7,11), 2 and 11 (2 primes) divide 22, but 3 and 5 and 7 (3 primes) do not divide 22.
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A111905, A111907.

Programs

PARI

{m=115;v=vector(m);for(n=2,m,f=factor(n)[,1]~;c=0;pc=0;forprime(p=2,vecmax(f), j=1;s=length(f);while(j<=s&&p!=f[j],j++);if(j<=s,c++);pc++);v[n]=sign(pc-2*c)); for(n=1,m,if(v[n]>0,print1(n,",")))} \\ Klaus Brockhaus, Aug 21 2005

Python

from itertools import count, islice
from sympy import sieve, factorint
def a_gen():
    for n in count(3):
        f = [sieve.search(i)[0] for i in factorint(n)]
        if len(f) < (f[-1]+1)//2:
            yield n
A111906_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Jun 22 2024

Extensions

More terms from Klaus Brockhaus, Aug 21 2005