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A190106 Numbers with prime factorization p^2*q^3*r^3 where p, q, and r are distinct primes.

%I A190106 #26 Apr 22 2025 04:33:12
%S A190106 5400,9000,10584,13500,24696,26136,36504,37044,49000,62424,68600,
%T A190106 77976,95832,114264,121000,143748,158184,165375,169000,171500,181656,
%U A190106 207576,231525,237276,266200,289000,295704,332024,353736,361000,363096
%N A190106 Numbers with prime factorization p^2*q^3*r^3 where p, q, and r are distinct primes.
%H A190106 T. D. Noe, <a href="/A190106/b190106.txt">Table of n, a(n) for n = 1..1000</a>
%H A190106 Will Nicholes, <a href="https://willnicholes.com/2010/06/06/list-of-prime-signatures">List of prime signatures</a>, 2010.
%H A190106 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>.
%F A190106 Sum_{n>=1} 1/a(n) = P(2)*P(3)^2/2 - P(2)*P(6)/2 - P(3)*P(5) + P(8) = 0.00085907862422456410530..., where P is the prime zeta function. - _Amiram Eldar_, Mar 07 2024
%t A190106 f[n_]:=Sort[Last/@FactorInteger[n]]=={2,3,3};Select[Range[500000],f]
%o A190106 (PARI) list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\4)^(1/6), t1=p^3;forprime(q=p+1, (lim\t1)^(1/3), t2=t1*q^3;forprime(r=2, sqrt(lim\t2), if(p==r||q==r, next);listput(v,t2*r^2)))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 20 2011
%o A190106 (Python)
%o A190106 from math import isqrt
%o A190106 from sympy import primepi, integer_nthroot, primerange
%o A190106 def A190106(n):
%o A190106     def bisection(f,kmin=0,kmax=1):
%o A190106         while f(kmax) > kmax: kmax <<= 1
%o A190106         kmin = kmax >> 1
%o A190106         while kmax-kmin > 1:
%o A190106             kmid = kmax+kmin>>1
%o A190106             if f(kmid) <= kmid:
%o A190106                 kmax = kmid
%o A190106             else:
%o A190106                 kmin = kmid
%o A190106         return kmax
%o A190106     def f(x): return n+x+sum((t:=primepi(s:=isqrt(y:=integer_nthroot(x//r**2,3)[0])))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(isqrt(x)+1))+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 A190106     return bisection(f,n,n) # _Chai Wah Wu_, Mar 27 2025
%Y A190106 Cf. A179691, A179698, A179746, A189991.
%Y A190106 Cf. A085548, A085541, A085965, A085966, A085968.
%K A190106 nonn
%O A190106 1,1
%A A190106 _Vladimir Joseph Stephan Orlovsky_, May 04 2011