A030627 Numbers with 9 divisors.

36, 100, 196, 225, 256, 441, 484, 676, 1089, 1156, 1225, 1444, 1521, 2116, 2601, 3025, 3249, 3364, 3844, 4225, 4761, 5476, 5929, 6561, 6724, 7225, 7396, 7569, 8281, 8649, 8836, 9025, 11236, 12321, 13225, 13924, 14161, 14884, 15129

Offset: 1

Author

Jeff Burch

Keywords

• nonn

Comments

Numbers of the form p^8 (8th row of A120458) or p^2*r^2 (A085986), where p and r are distinct primes. - R. J. Mathar, Mar 01 2010

Links

• R. J. Mathar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000005, A030515, A030516, A030626, A085986, A120458, A179645.

Programs

Mathematica

Select[Range[90000],DivisorSigma[0,#]==9&] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

PARI

is(n)=numdiv(n)==9 \\ Charles R Greathouse IV, Jun 19 2016

Python

from math import isqrt
from sympy import primepi, primerange
def A030627(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x+(t:=primepi(s:=isqrt(y:=isqrt(x))))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1))-primepi(isqrt(s)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Formula

A000005(a(n)) = 9. - Juri-Stepan Gerasimov, Oct 10 2009

Sum_{n>=1} 1/a(n) = (P(2)^2 - P(4))/2 + P(8) = 0.0678286..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022