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A189990 Numbers with prime factorization p^2*q^6.

%I A189990 #37 Mar 28 2025 03:27:31
%S A189990 576,1600,2916,3136,7744,10816,18225,18496,23104,33856,35721,53824,
%T A189990 61504,62500,87616,88209,107584,118336,123201,140625,141376,179776,
%U A189990 210681,222784,238144,263169,287296,322624,341056,385641,399424,440896,470596
%N A189990 Numbers with prime factorization p^2*q^6.
%H A189990 T. D. Noe, <a href="/A189990/b189990.txt">Table of n, a(n) for n = 1..1000</a>
%H A189990 Will Nicholes, <a href="https://willnicholes.com/2010/06/06/list-of-prime-signatures">List of Prime Signatures</a>
%H A189990 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F A189990 Sum_{n>=1} 1/a(n) = P(2)*P(6) - P(8) = A085548 * A085966 - A085968 = 0.003658..., where P is the prime zeta function. - _Amiram Eldar_, Jul 06 2020
%F A189990 a(n) = A065036(n)^2. - _Chai Wah Wu_, Mar 27 2025
%t A189990 f[n_]:=Sort[Last/@FactorInteger[n]]=={2,6}; Select[Range[800000],f]
%o A189990 (PARI) list(lim)=my(v=List(),t);forprime(p=2, (lim\4)^(1/6), t=p^6;forprime(q=2, sqrt(lim\t), if(p==q, next);listput(v,t*q^2))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 24 2011
%o A189990 (Python)
%o A189990 from math import isqrt
%o A189990 from sympy import primepi, integer_nthroot, primerange
%o A189990 def A189990(n):
%o A189990     def bisection(f,kmin=0,kmax=1):
%o A189990         while f(kmax) > kmax: kmax <<= 1
%o A189990         kmin = kmax >> 1
%o A189990         while kmax-kmin > 1:
%o A189990             kmid = kmax+kmin>>1
%o A189990             if f(kmid) <= kmid:
%o A189990                 kmax = kmid
%o A189990             else:
%o A189990                 kmin = kmid
%o A189990         return kmax
%o A189990     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])
%o A189990     return bisection(f,n,n) # _Chai Wah Wu_, Feb 21 2025
%Y A189990 Subsequence of A137484.
%Y A189990 Cf. A179645, A179664.
%Y A189990 Cf. A065036, A085548, A085966, A085968.
%K A189990 nonn
%O A189990 1,1
%A A189990 _Vladimir Joseph Stephan Orlovsky_, May 03 2011