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A175745 Numbers with 35 divisors.

%I A175745 #27 Feb 22 2025 11:21:18
%S A175745 5184,11664,40000,153664,250000,455625,937024,1265625,1750329,1827904,
%T A175745 1882384,5345344,8340544,9529569,10673289,17909824,20820969,28344976,
%U A175745 37515625,45265984,59105344,60886809,73530625,77228944,95004009,119946304,143496441,180848704
%N A175745 Numbers with 35 divisors.
%C A175745 Numbers of the form p^34 and p^6*q^4 (A190464), where p and q are distinct primes.
%H A175745 T. D. Noe, <a href="/A175745/b175745.txt">Table of n, a(n) for n = 1..1000</a>
%H A175745 OEIS Wiki, <a href="https://oeis.org/wiki/Index_entries_for_number_of_divisors">Index entries for number of divisors</a>
%F A175745 A000005(a(n)) = 35.
%F A175745 Sum_{n>=1} 1/a(n) = P(4)*P(6) - P(10) + P(34) = 0.000320676..., where P is the prime zeta function. - _Amiram Eldar_, Jul 03 2022
%t A175745 Select[Range[9000000],DivisorSigma[0,#]==35&] (* _Vladimir Joseph Stephan Orlovsky_, May 06 2011 *)
%o A175745 (PARI) is(n)=numdiv(n)==35 \\ _Charles R Greathouse IV_, Jun 19 2016
%o A175745 (Python)
%o A175745 from sympy import primepi, integer_nthroot, primerange
%o A175745 def A175745(n):
%o A175745     def bisection(f,kmin=0,kmax=1):
%o A175745         while f(kmax) > kmax: kmax <<= 1
%o A175745         kmin = kmax >> 1
%o A175745         while kmax-kmin > 1:
%o A175745             kmid = kmax+kmin>>1
%o A175745             if f(kmid) <= kmid:
%o A175745                 kmax = kmid
%o A175745             else:
%o A175745                 kmin = kmid
%o A175745         return kmax
%o A175745     def f(x): return n+x-sum(primepi(integer_nthroot(x//p**6,4)[0]) for p in primerange(integer_nthroot(x,6)[0]+1))+primepi(integer_nthroot(x,10)[0])-primepi(integer_nthroot(x,34)[0])
%o A175745     return bisection(f,n,n) # _Chai Wah Wu_, Feb 22 2025
%Y A175745 Cf. A000005, A175742, A175743, A175744.
%K A175745 nonn
%O A175745 1,1
%A A175745 _Jaroslav Krizek_, Aug 27 2010
%E A175745 Extended by _T. D. Noe_, May 08 2011