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A179669 Products of form p^4*q^2*r where p, q and r are three distinct primes.

%I A179669 #30 Apr 22 2025 04:11:45
%S A179669 720,1008,1200,1584,1620,1872,2268,2352,2448,2736,2800,3312,3564,3920,
%T A179669 4050,4176,4212,4400,4464,5200,5328,5508,5808,5904,6156,6192,6768,
%U A179669 6800,7452,7500,7600,7632,7938,8112,8496,8624,8784,9200,9396,9648,9680,10044
%N A179669 Products of form p^4*q^2*r where p, q and r are three distinct primes.
%H A179669 T. D. Noe, <a href="/A179669/b179669.txt">Table of n, a(n) for n = 1..1000</a>
%H A179669 Will Nicholes, <a href="https://willnicholes.com/2010/06/06/list-of-prime-signatures">List of Prime Signatures</a>
%H A179669 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%p A179669 N:= 20000: # for terms < N
%p A179669 P:= select(isprime,[2,seq(i,i=1..N/(3^2*2^4),2)]):
%p A179669 R:= NULL:
%p A179669 for i from 1 while P[i]^4 * 2^2*3 < N do
%p A179669   for j from 1 while P[i]^4 * P[j]^2 *2 < N do
%p A179669     if j = i then next fi;
%p A179669     m:= ListTools:-BinaryPlace(P,N/P[i]^4/P[j]^2);
%p A179669     R:= R, seq(P[i]^4*P[j]^2*P[k],k={$1..m} minus {i,j});
%p A179669 od od:
%p A179669 sort([R]); # _Robert Israel_, Mar 28 2025
%t A179669 f[n_]:=Sort[Last/@FactorInteger[n]]=={1,2,4}; Select[Range[10000], f]
%o A179669 (PARI) list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\12)^(1/4), t1=p^4;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 A179669 (Python)
%o A179669 from math import isqrt
%o A179669 from sympy import primepi, primerange, integer_nthroot
%o A179669 def A179669(n):
%o A179669     def bisection(f,kmin=0,kmax=1):
%o A179669         while f(kmax) > kmax: kmax <<= 1
%o A179669         kmin = kmax >> 1
%o A179669         while kmax-kmin > 1:
%o A179669             kmid = kmax+kmin>>1
%o A179669             if f(kmid) <= kmid:
%o A179669                 kmax = kmid
%o A179669             else:
%o A179669                 kmin = kmid
%o A179669         return kmax
%o A179669     def f(x): return n+x-sum(primepi(x//(p**4*q**2)) for p in primerange(integer_nthroot(x,4)[0]+1) for q in primerange(isqrt(x//p**4)+1))+sum(primepi(integer_nthroot(x//p**4,3)[0]) for p in primerange(integer_nthroot(x,4)[0]+1))+sum(primepi(isqrt(x//p**5)) for p in primerange(integer_nthroot(x,5)[0]+1))+sum(primepi(x//p**6) for p in primerange(integer_nthroot(x,6)[0]+1))-(primepi(integer_nthroot(x,7)[0])<<1)
%o A179669     return bisection(f,n,n) # _Chai Wah Wu_, Mar 28 2025
%Y A179669 Cf. A137493.
%K A179669 nonn
%O A179669 1,1
%A A179669 _Vladimir Joseph Stephan Orlovsky_, Jul 23 2010