A137484 Numbers with 21 divisors.

576, 1600, 2916, 3136, 7744, 10816, 18225, 18496, 23104, 33856, 35721, 53824, 61504, 62500, 87616, 88209, 107584, 118336, 123201, 140625, 141376, 179776, 210681, 222784, 238144, 263169, 287296, 322624, 341056, 385641, 399424, 440896

Offset: 1

Author

R. J. Mathar, Apr 22 2008

Keywords

• nonn

Comments

Maple implementation: see A030513.

Numbers of the form p^20 or p^2*q^6 (A189990) where p and q are distinct primes. - R. J. Mathar, Mar 01 2010

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000005, A030513, A030638 (20 divisors), A137485 (22 divisors), A189990.

Programs

Mathematica

Select[Range[450000],DivisorSigma[0,#]==21&] (* Vladimir Joseph Stephan Orlovsky, May 03 2011 *)

PARI

is(n)=numdiv(n)==21 \\ Charles R Greathouse IV, Jun 19 2016

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A137484(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 n+x-sum(primepi(isqrt(x//p**6)) for p in primerange(integer_nthroot(x,6)[0]+1))+primepi(integer_nthroot(x,8)[0])-primepi(integer_nthroot(x,20)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Formula

A000005(a(n)) = 21.

Sum_{n>=1} 1/a(n) = P(2)*P(6) - P(8) + P(20) = 0.00365945..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022