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A190115 Numbers with prime factorization p^2*q^3*r^4 where p, q, and r are distinct primes.

%I A190115 #25 Apr 22 2025 04:33:16
%S A190115 10800,16200,18000,21168,31752,40500,45000,49392,52272,67500,73008,
%T A190115 78408,98000,109512,111132,124848,137200,155952,172872,187272,191664,
%U A190115 228528,233928,242000,245000,259308,316368,338000,342792,363312,415152
%N A190115 Numbers with prime factorization p^2*q^3*r^4 where p, q, and r are distinct primes.
%H A190115 T. D. Noe, <a href="/A190115/b190115.txt">Table of n, a(n) for n = 1..1000</a>
%H A190115 Will Nicholes, <a href="https://willnicholes.com/2010/06/06/list-of-prime-signatures">List of prime signatures</a>, 2010.
%H A190115 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>.
%F A190115 Sum_{n>=1} 1/a(n) = P(2)*P(3)*P(4) - P(2)*P(7) - P(3)*P(6) - P(4)*P(5) + 2*P(9) = 0.00061171477910848082277..., where P is the prime zeta function. - _Amiram Eldar_, Mar 07 2024
%t A190115 f[n_]:=Sort[Last/@FactorInteger[n]]=={2,3,4};Select[Range[900000],f]
%o A190115 (PARI) list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\72)^(1/4), t1=p^4;forprime(q=2, (lim\t1)^(1/3), if(p==q, next);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 24 2011
%o A190115 (Python)
%o A190115 from math import isqrt
%o A190115 from sympy import primepi, primerange, integer_nthroot
%o A190115 def A190115(n):
%o A190115     def bisection(f,kmin=0,kmax=1):
%o A190115         while f(kmax) > kmax: kmax <<= 1
%o A190115         kmin = kmax >> 1
%o A190115         while kmax-kmin > 1:
%o A190115             kmid = kmax+kmin>>1
%o A190115             if f(kmid) <= kmid:
%o A190115                 kmax = kmid
%o A190115             else:
%o A190115                 kmin = kmid
%o A190115         return kmax
%o A190115     def f(x): return n+x-sum(primepi(isqrt(x//(r**4*q**3))) for r in primerange(integer_nthroot(x,4)[0]+1) for q in primerange(integer_nthroot(x//r**4,3)[0]+1))+sum(primepi(integer_nthroot(x//p**5,4)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))+sum(primepi(integer_nthroot(x//p**6,3)[0]) for p in primerange(integer_nthroot(x,6)[0]+1))+sum(primepi(isqrt(x//p**7)) for p in primerange(integer_nthroot(x,7)[0]+1))-(primepi(integer_nthroot(x,9)[0])<<1)
%o A190115     return bisection(f,n,n) # _Chai Wah Wu_, Mar 27 2025
%Y A190115 Cf. A093770, A190012, A190014.
%Y A190115 Cf. A085548, A085541, A085964, A085965, A085966, A085967, A085969.
%K A190115 nonn
%O A190115 1,1
%A A190115 _Vladimir Joseph Stephan Orlovsky_, May 04 2011