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A179671 Products of the 5th power of a prime and a distinct prime of the 3rd power (p^5*q^3).

%I A179671 #19 Feb 21 2025 19:53:29
%S A179671 864,1944,4000,10976,25000,30375,42592,70304,83349,84375,134456,
%T A179671 157216,219488,323433,389344,453789,533871,780448,953312,1071875,
%U A179671 1193859,1288408,1620896,1666737,2100875,2205472,2544224,2956581,2970344,3322336,4159375,4348377
%N A179671 Products of the 5th power of a prime and a distinct prime of the 3rd power (p^5*q^3).
%H A179671 T. D. Noe, <a href="/A179671/b179671.txt">Table of n, a(n) for n = 1..1000</a>
%H A179671 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F A179671 Sum_{n>=1} 1/a(n) = P(3)*P(5) - P(8) = A085541 * A085965 - A085968 = 0.002187..., where P is the prime zeta function. - _Amiram Eldar_, Jul 06 2020
%t A179671 f[n_]:=Sort[Last/@FactorInteger[n]]=={3,5}; Select[Range[10^6], f]
%o A179671 (Python)
%o A179671 from sympy import primepi, integer_nthroot, primerange
%o A179671 def A179671(n):
%o A179671     def bisection(f,kmin=0,kmax=1):
%o A179671         while f(kmax) > kmax: kmax <<= 1
%o A179671         kmin = kmax >> 1
%o A179671         while kmax-kmin > 1:
%o A179671             kmid = kmax+kmin>>1
%o A179671             if f(kmid) <= kmid:
%o A179671                 kmax = kmid
%o A179671             else:
%o A179671                 kmin = kmid
%o A179671         return kmax
%o A179671     def f(x): return n+x-sum(primepi(integer_nthroot(x//p**5,3)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))+primepi(integer_nthroot(x,8)[0])
%o A179671     return bisection(f,n,n) # _Chai Wah Wu_, Feb 21 2025
%Y A179671 Cf. A085541, A085965, A085968.
%K A179671 nonn
%O A179671 1,1
%A A179671 _Vladimir Joseph Stephan Orlovsky_, Jul 23 2010