cp's OEIS Frontend

A179694 Numbers of the form p^6*q^3 where p and q are distinct primes.

%I A179694 #18 Feb 21 2025 19:39:29
%S A179694 1728,5832,8000,21952,85184,91125,125000,140608,250047,314432,421875,
%T A179694 438976,778688,941192,970299,1560896,1601613,1906624,3176523,3241792,
%U A179694 3581577,4410944,5000211,5088448,5359375,6644672
%N A179694 Numbers of the form p^6*q^3 where p and q are distinct primes.
%H A179694 T. D. Noe, <a href="/A179694/b179694.txt">Table of n, a(n) for n = 1..1000</a>
%H A179694 Will Nicholes, <a href="http://willnicholes.com/math/primesiglist.htm">Prime Signatures</a>
%F A179694 Sum_{n>=1} 1/a(n) = P(3)*P(6) - P(9) = A085541 * A085966 - A085969 = 0.000978..., where P is the prime zeta function.  - _Amiram Eldar_, Jul 06 2020
%F A179694 a(n) = A054753(n)^3. - _R. J. Mathar_, May 05 2023
%t A179694 f[n_]:=Sort[Last/@FactorInteger[n]]=={3,6}; Select[Range[10^6], f]
%o A179694 (PARI) list(lim)=my(v=List(),t);forprime(p=2, (lim\8)^(1/6), t=p^6;forprime(q=2, (lim\t)^(1/3), if(p==q, next);listput(v,t*q^3))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 24 2011
%o A179694 (Python)
%o A179694 from sympy import primepi, integer_nthroot, primerange
%o A179694 def A179694(n):
%o A179694     def bisection(f,kmin=0,kmax=1):
%o A179694         while f(kmax) > kmax: kmax <<= 1
%o A179694         kmin = kmax >> 1
%o A179694         while kmax-kmin > 1:
%o A179694             kmid = kmax+kmin>>1
%o A179694             if f(kmid) <= kmid:
%o A179694                 kmax = kmid
%o A179694             else:
%o A179694                 kmin = kmid
%o A179694         return kmax
%o A179694     def f(x): return n+x-sum(primepi(integer_nthroot(x//p**6,3)[0]) for p in primerange(integer_nthroot(x,6)[0]+1))+primepi(integer_nthroot(x,9)[0])
%o A179694     return bisection(f,n,n) # _Chai Wah Wu_, Feb 21 2025
%Y A179694 Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688, A179689, A179690, A179691, A179692, A179693.
%Y A179694 Cf. A085541, A085966, A085969.
%K A179694 nonn
%O A179694 1,1
%A A179694 _Vladimir Joseph Stephan Orlovsky_, Jul 24 2010