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A190377 Numbers with prime factorization p^2*q^2*r^2*s^2 where p, q, r, and s are distinct primes.

%I A190377 #24 Apr 22 2025 04:33:19
%S A190377 44100,108900,152100,213444,260100,298116,324900,476100,509796,592900,
%T A190377 636804,736164,756900,828100,864900,933156,1232100,1258884,1334025,
%U A190377 1416100,1483524,1512900,1572516,1664100,1695204,1758276,1768900,1863225
%N A190377 Numbers with prime factorization p^2*q^2*r^2*s^2 where p, q, r, and s are distinct primes.
%H A190377 T. D. Noe, <a href="/A190377/b190377.txt">Table of n, a(n) for n = 1..1000</a>
%H A190377 Will Nicholes, <a href="https://willnicholes.com/2010/06/06/list-of-prime-signatures">List of prime signatures</a>, 2010.
%H A190377 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>.
%F A190377 Sum_{n>=1} 1/a(n) = (P(2)^4 - 6*P(2)^2*P(4) + 8*P(2)*P(6) + 3*P(4)^2 - 6*P(8))/24 = 0.00010511750849230980748..., where P is the prime zeta function. - _Amiram Eldar_, Mar 07 2024
%F A190377 a(n) = A046386(n)^2. - _Chai Wah Wu_, Mar 27 2025
%t A190377 f[n_]:=Sort[Last/@FactorInteger[n]]=={2,2,2,2};Select[Range[3000000],f]
%o A190377 (PARI) list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtint(lim\900), t1=p^2; forprime(q=2,sqrtint(lim\(36*t1)), if(q==p, next); t2=q^2*t1; forprime(r=2,sqrtint(lim\(4*t2)), if(r==p || r==q, next); t3=r^2*t2; forprime(s=2,sqrtint(lim\t3), if(s==p || s==q || s==r, next); listput(v, t3*s^2))))); Set(v) \\ _Charles R Greathouse IV_, Aug 25 2016
%o A190377 (Python)
%o A190377 from math import isqrt
%o A190377 from sympy import primepi, primerange, integer_nthroot
%o A190377 def A190377(n):
%o A190377     def bisection(f,kmin=0,kmax=1):
%o A190377         while f(kmax) > kmax: kmax <<= 1
%o A190377         kmin = kmax >> 1
%o A190377         while kmax-kmin > 1:
%o A190377             kmid = kmax+kmin>>1
%o A190377             if f(kmid) <= kmid:
%o A190377                 kmax = kmid
%o A190377             else:
%o A190377                 kmin = kmid
%o A190377         return kmax
%o A190377     def f(x): return int(n+x-sum(primepi(x//(k*m*r))-c for a,k in enumerate(primerange(integer_nthroot(x,4)[0]+1),1) for b,m in enumerate(primerange(k+1,integer_nthroot(x//k,3)[0]+1),a+1) for c,r in enumerate(primerange(m+1,isqrt(x//(k*m))+1),b+1)))
%o A190377     return bisection(f,n,n)**2 # _Chai Wah Wu_, Mar 27 2025
%Y A190377 Cf. A190108, A190109.
%Y A190377 Cf. A046386, A085548, A085964, A085966, A085968.
%K A190377 nonn
%O A190377 1,1
%A A190377 _Vladimir Joseph Stephan Orlovsky_, May 09 2011