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A179695 Numbers of the form p^3*q^2*r^2 where p, q, and r are distinct primes.

%I A179695 #34 Apr 22 2025 04:16:16
%S A179695 1800,2700,3528,4500,5292,8712,9800,12168,12348,13068,18252,20808,
%T A179695 24200,24500,25992,31212,33075,33800,34300,38088,38988,47432,47916,
%U A179695 55125,57132,57800,60500,60552,66248,69192,72200,77175,79092,81675,84500
%N A179695 Numbers of the form p^3*q^2*r^2 where p, q, and r are distinct primes.
%C A179695 Subsequence of A225228. - _Reinhard Zumkeller_, May 03 2013
%H A179695 T. D. Noe, <a href="/A179695/b179695.txt">Table of n, a(n) for n = 1..1000</a>
%H A179695 Will Nicholes, <a href="https://willnicholes.com/2010/06/06/list-of-prime-signatures">List of prime signatures</a>, 2010.
%H A179695 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>.
%F A179695 A050326(a(n)) = 5. - _Reinhard Zumkeller_, May 03 2013
%F A179695 Sum_{n>=1} 1/a(n) = P(2)^2*P(3)/2 - P(3)*P(4)/2 - P(2)*P(5) + P(7) = 0.0032578591481263202818..., where P is the prime zeta function. - _Amiram Eldar_, Mar 07 2024
%t A179695 f[n_]:=Sort[Last/@FactorInteger[n]]=={2,2,3}; Select[Range[10^5], f]
%t A179695 f[n_]:={Times@@(n^{2,2,3}),Times@@(n^{2,3,2}),Times@@(n^{3,2,2})}; Module[ {nn=20},Select[Flatten[f/@Subsets[Prime[Range[nn]],{3}]],#<= 72*Prime[ nn]^2&]]//Union (* _Harvey P. Dale_, Jul 05 2019 *)
%o A179695 (PARI) list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\36)^(1/3), t1=p^3;forprime(q=2, sqrt(lim\t1), if(p==q, next);t2=t1*q^2;forprime(r=q+1, sqrt(lim\t2), if(p==r,next);listput(v,t2*r^2)))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 24 2011
%o A179695 (Python)
%o A179695 from math import isqrt
%o A179695 from sympy import primepi, primerange, integer_nthroot
%o A179695 def A179695(n):
%o A179695     def bisection(f,kmin=0,kmax=1):
%o A179695         while f(kmax) > kmax: kmax <<= 1
%o A179695         kmin = kmax >> 1
%o A179695         while kmax-kmin > 1:
%o A179695             kmid = kmax+kmin>>1
%o A179695             if f(kmid) <= kmid:
%o A179695                 kmax = kmid
%o A179695             else:
%o A179695                 kmin = kmid
%o A179695         return kmax
%o A179695     def f(x): return n+x+sum((t:=primepi(s:=isqrt(y:=isqrt(x//r**3))))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(integer_nthroot(x,3)[0]+1))+sum(primepi(isqrt(x//p**5)) for p in primerange(integer_nthroot(x,5)[0]+1))-primepi(integer_nthroot(x,7)[0])
%o A179695     return bisection(f,n,n) # _Chai Wah Wu_, Mar 27 2025
%Y A179695 Cf. A050326, A225228.
%Y A179695 Cf. A085548, A085541, A085964, A085965, A085967.
%K A179695 nonn
%O A179695 1,1
%A A179695 _Vladimir Joseph Stephan Orlovsky_, Jul 24 2010