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

1,1

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

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A235181 A268773 A250789 * A189990 A064254 A281242

Adjacent sequences: A137481 A137482 A137483 * A137485 A137486 A137487

Keyword

nonn,changed

Author

R. J. Mathar, Apr 22 2008

Status

approved