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A137484 Numbers with 21 divisors.

%I A137484 #24 Feb 22 2025 01:22:57
%S A137484 576,1600,2916,3136,7744,10816,18225,18496,23104,33856,35721,53824,
%T A137484 61504,62500,87616,88209,107584,118336,123201,140625,141376,179776,
%U A137484 210681,222784,238144,263169,287296,322624,341056,385641,399424,440896
%N A137484 Numbers with 21 divisors.
%C A137484 Maple implementation: see A030513.
%C A137484 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
%H A137484 T. D. Noe, <a href="/A137484/b137484.txt">Table of n, a(n) for n = 1..1000</a>
%F A137484 A000005(a(n)) = 21.
%F A137484 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
%t A137484 Select[Range[450000],DivisorSigma[0,#]==21&] (* _Vladimir Joseph Stephan Orlovsky_, May 03 2011 *)
%o A137484 (PARI) is(n)=numdiv(n)==21 \\ _Charles R Greathouse IV_, Jun 19 2016
%o A137484 (Python)
%o A137484 from math import isqrt
%o A137484 from sympy import primepi, primerange, integer_nthroot
%o A137484 def A137484(n):
%o A137484     def bisection(f,kmin=0,kmax=1):
%o A137484         while f(kmax) > kmax: kmax <<= 1
%o A137484         kmin = kmax >> 1
%o A137484         while kmax-kmin > 1:
%o A137484             kmid = kmax+kmin>>1
%o A137484             if f(kmid) <= kmid:
%o A137484                 kmax = kmid
%o A137484             else:
%o A137484                 kmin = kmid
%o A137484         return kmax
%o A137484     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])
%o A137484     return bisection(f,n,n) # _Chai Wah Wu_, Feb 21 2025
%Y A137484 Cf. A000005, A030513, A030638 (20 divisors), A137485 (22 divisors), A189990.
%K A137484 nonn
%O A137484 1,1
%A A137484 _R. J. Mathar_, Apr 22 2008