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A163569 Numbers of the form p^3*q^2*r where p, q and r are three distinct primes.

%I A163569 #21 Apr 22 2025 03:59:46
%S A163569 360,504,540,600,756,792,936,1176,1188,1224,1350,1368,1400,1404,1500,
%T A163569 1656,1836,1960,2052,2088,2200,2232,2250,2484,2600,2646,2664,2904,
%U A163569 2952,3096,3132,3348,3384,3400,3500,3800,3816,3996,4056,4116,4248,4312,4392
%N A163569 Numbers of the form p^3*q^2*r where p, q and r are three distinct primes.
%C A163569 There is no constraint on which of the three primes is the largest or smallest.
%H A163569 T. D. Noe, <a href="/A163569/b163569.txt">Table of n, a(n) for n = 1..1000</a>
%H A163569 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%e A163569 360=2^3*3^2*5. 504=2^3*3^2*7. 1188=2^2*3^3*11.
%t A163569 f[n_]:=Sort[Last/@FactorInteger[n]]=={1,2,3}; Select[Range[5000], f]
%o A163569 (PARI) list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\12)^(1/3), t1=p^3;forprime(q=2, sqrt(lim\t1), if(p==q, next);t2=t1*q^2;forprime(r=2, lim\t2, if(p==r||q==r, next);listput(v,t2*r)))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 20 2011
%o A163569 (Python)
%o A163569 from math import isqrt
%o A163569 from sympy import primepi, primerange, integer_nthroot
%o A163569 def A163569(n):
%o A163569     def bisection(f,kmin=0,kmax=1):
%o A163569         while f(kmax) > kmax: kmax <<= 1
%o A163569         kmin = kmax >> 1
%o A163569         while kmax-kmin > 1:
%o A163569             kmid = kmax+kmin>>1
%o A163569             if f(kmid) <= kmid:
%o A163569                 kmax = kmid
%o A163569             else:
%o A163569                 kmin = kmid
%o A163569         return kmax
%o A163569     def f(x): return n+x-sum(primepi(x//(p**3*q**2)) for p in primerange(integer_nthroot(x,3)[0]+1) for q in primerange(isqrt(x//p**3)+1))+sum(primepi(integer_nthroot(x//p**3,3)[0]) for p in primerange(integer_nthroot(x,3)[0]+1))+sum(primepi(isqrt(x//p**4)) for p in primerange(integer_nthroot(x,4)[0]+1))+sum(primepi(x//p**5) for p in primerange(integer_nthroot(x,5)[0]+1))-(primepi(integer_nthroot(x,6)[0])<<1)
%o A163569     return bisection(f,n,n) # _Chai Wah Wu_, Mar 27 2025
%Y A163569 Subsequence of A137487. - _R. J. Mathar_, Aug 01 2009
%K A163569 nonn
%O A163569 1,1
%A A163569 _Vladimir Joseph Stephan Orlovsky_, Jul 31 2009