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A030632 Numbers with 14 divisors.

%I A030632 #29 Feb 22 2025 01:22:40
%S A030632 192,320,448,704,832,1088,1216,1458,1472,1856,1984,2368,2624,2752,
%T A030632 3008,3392,3645,3776,3904,4288,4544,4672,5056,5103,5312,5696,6208,
%U A030632 6464,6592,6848,6976,7232,8019,8128,8192,8384,8768,8896,9477,9536,9664,10048,10432
%N A030632 Numbers with 14 divisors.
%C A030632 Numbers of the form p^13 (A138031) or p*q^6 (A189987), where p and q are distinct primes. - _R. J. Mathar_, Mar 01 2010
%H A030632 R. J. Mathar, <a href="/A030632/b030632.txt">Table of n, a(n) for n = 1..1000</a>
%t A030632 Select[Range[15000], DivisorSigma[0, #] == 14 &]
%o A030632 (PARI) is(n)=numdiv(n)==14 \\ _Charles R Greathouse IV_, Jun 19 2016
%o A030632 (Python)
%o A030632 from sympy import primepi, primerange, integer_nthroot
%o A030632 def A030632(n):
%o A030632     def bisection(f,kmin=0,kmax=1):
%o A030632         while f(kmax) > kmax: kmax <<= 1
%o A030632         kmin = kmax >> 1
%o A030632         while kmax-kmin > 1:
%o A030632             kmid = kmax+kmin>>1
%o A030632             if f(kmid) <= kmid:
%o A030632                 kmax = kmid
%o A030632             else:
%o A030632                 kmin = kmid
%o A030632         return kmax
%o A030632     def f(x): return n+x-sum(primepi(x//p**6) for p in primerange(integer_nthroot(x,6)[0]+1))+primepi(integer_nthroot(x,7)[0])-primepi(integer_nthroot(x,13)[0])
%o A030632     return bisection(f,n,n) # _Chai Wah Wu_, Feb 22 2025
%Y A030632 Cf. A092759.
%K A030632 nonn
%O A030632 1,1
%A A030632 _Jeff Burch_