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A030633 Numbers with 15 divisors.

%I A030633 #28 Feb 22 2025 01:22:47
%S A030633 144,324,400,784,1936,2025,2500,2704,3969,4624,5625,5776,8464,9604,
%T A030633 9801,13456,13689,15376,16384,21609,21904,23409,26896,29241,29584,
%U A030633 30625,35344,42849,44944,55696,58564,59536,60025,68121,71824,75625
%N A030633 Numbers with 15 divisors.
%C A030633 Numbers of the form p^14 (subset of A010802) or p^2*q^4 (A189988) where p and q are distinct primes. - _R. J. Mathar_, Mar 01 2010
%H A030633 R. J. Mathar, <a href="/A030633/b030633.txt">Table of n, a(n) for n = 1..1000</a>
%F A030633 From _Amiram Eldar_, Jul 03 2022: (Start)
%F A030633 A000005(a(n)) = 15.
%F A030633 Sum_{n>=1} 1/a(n) = P(2)*P(4) - P(6) + P(14) = 0.0178111..., where P is the prime zeta function. (End)
%t A030633 Select[Range[300000],DivisorSigma[0,#]==15&] (* _Vladimir Joseph Stephan Orlovsky_, May 05 2011 *)
%o A030633 (PARI) is(n)=numdiv(n)==15 \\ _Charles R Greathouse IV_, Jun 19 2016
%o A030633 (Python)
%o A030633 from math import isqrt
%o A030633 from sympy import primepi, primerange, integer_nthroot
%o A030633 def A030633(n):
%o A030633     def bisection(f,kmin=0,kmax=1):
%o A030633         while f(kmax) > kmax: kmax <<= 1
%o A030633         kmin = kmax >> 1
%o A030633         while kmax-kmin > 1:
%o A030633             kmid = kmax+kmin>>1
%o A030633             if f(kmid) <= kmid:
%o A030633                 kmax = kmid
%o A030633             else:
%o A030633                 kmin = kmid
%o A030633         return kmax
%o A030633     def f(x): return n+x-sum(primepi(isqrt(x//p**4)) for p in primerange(integer_nthroot(x,4)[0]+1))+primepi(integer_nthroot(x,6)[0])-primepi(integer_nthroot(x,14)[0])
%o A030633     return bisection(f,n,n) # _Chai Wah Wu_, Feb 22 2025
%Y A030633 Cf. A000005, A010802, A030630, A030631, A030632, A189988.
%K A030633 nonn
%O A030633 1,1
%A A030633 _Jeff Burch_