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A179646 Product of the 5th power of a prime and different distinct prime of the 2nd power (p^5*q^2).

%I A179646 #26 Feb 21 2025 21:44:22
%S A179646 288,800,972,1568,3872,5408,6075,9248,11552,11907,12500,16928,26912,
%T A179646 28125,29403,30752,41067,43808,53792,59168,67228,70227,70688,87723,
%U A179646 89888,111392,119072,128547,143648,151263,153125,161312,170528,199712
%N A179646 Product of the 5th power of a prime and different distinct prime of the 2nd power (p^5*q^2).
%C A179646 288=2^5*3^2, 800=2^5*5^2,..
%H A179646 T. D. Noe, <a href="/A179646/b179646.txt">Table of n, a(n) for n = 1..1000</a>
%H A179646 Will Nicholes, <a href="http://willnicholes.com/math/primesiglist.htm">Prime Signatures</a>
%H A179646 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F A179646 Sum_{n>=1} 1/a(n) = P(2)*P(5) - P(7) = A085548 * A085965 - A085967 = 0.007886..., where P is the prime zeta function. - _Amiram Eldar_, Jul 06 2020
%t A179646 f[n_]:=Sort[Last/@FactorInteger[n]]=={2,5}; Select[Range[200000], f]
%o A179646 (PARI) list(lim)=my(v=List(),t);forprime(p=2,(lim\4)^(1/5),t=p^5;forprime(q=2,sqrt(lim\t),if(p==q,next);listput(v,t*q^2)));vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 20 2011
%o A179646 (Python)
%o A179646 from math import isqrt
%o A179646 from sympy import primepi, primerange, integer_nthroot
%o A179646 def A189988(n):
%o A179646     def bisection(f,kmin=0,kmax=1):
%o A179646         while f(kmax) > kmax: kmax <<= 1
%o A179646         kmin = kmax >> 1
%o A179646         while kmax-kmin > 1:
%o A179646             kmid = kmax+kmin>>1
%o A179646             if f(kmid) <= kmid:
%o A179646                 kmax = kmid
%o A179646             else:
%o A179646                 kmin = kmid
%o A179646         return kmax
%o A179646     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])
%o A179646     return bisection(f,n,n) # _Chai Wah Wu_, Feb 21 2025
%Y A179646 Cf. A030636, A046308, A007774.
%Y A179646 Cf. A085548, A085965, A085967.
%K A179646 nonn
%O A179646 1,1
%A A179646 _Vladimir Joseph Stephan Orlovsky_, Jul 21 2010