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A046386 Products of exactly four distinct primes.

%I A046386 #44 Jan 05 2025 19:44:04
%S A046386 210,330,390,462,510,546,570,690,714,770,798,858,870,910,930,966,1110,
%T A046386 1122,1155,1190,1218,1230,1254,1290,1302,1326,1330,1365,1410,1430,
%U A046386 1482,1518,1554,1590,1610,1722,1770,1785,1794,1806,1830,1870,1914,1938,1974
%N A046386 Products of exactly four distinct primes.
%C A046386 A squarefree subsequence of A033993. Numbers like 420 = 2^2*3*5*7 with at least one prime exponent greater than 1 in the prime signature are excluded here. - _R. J. Mathar_, Apr 03 2011
%C A046386 Numbers such that omega(n) = bigomega(n) = 4. - _Michel Marcus_, Dec 15 2015
%H A046386 T. D. Noe, <a href="/A046386/b046386.txt">Table of n, a(n) for n = 1..10000</a>
%F A046386 Intersection of A014613 (product of 4 primes) and A033993 (divisible by 4 distinct primes). - _M. F. Hasler_, Mar 24 2022
%e A046386 210 = 2*3*5*7;
%e A046386 330 = 2*3*5*11;
%e A046386 390 = 2*3*5*13;
%e A046386 462 = 2*3*7*11.
%t A046386 fQ[n_] := Last /@ FactorInteger[n] == {1, 1, 1, 1}; Select[ Range[2000], fQ[ # ] &] (* _Robert G. Wilson v_, Aug 04 2005 *)
%t A046386 Select[Range[2000],PrimeNu[#]==PrimeOmega[#]==4&] (* _Harvey P. Dale_, Jan 05 2025 *)
%o A046386 (PARI) is(n)=factor(n)[,2]==[1,1,1,1]~ \\ _Charles R Greathouse IV_, Sep 17 2015
%o A046386 (PARI) is(n) = omega(n)==4 && bigomega(n)==4 \\ _Hugo Pfoertner_, Dec 18 2018
%o A046386 (PARI) list(lim)=my(v=List()); forprime(p=2,sqrtnint(lim\=1,4), forprime(q=p+1,sqrtnint(lim\p,3), forprime(r=q+2,sqrtint(lim\p\q), my(t=p*q*r); forprime(s=r+2,lim\t, listput(v,t*s))))); Set(v) \\ _Charles R Greathouse IV_, Dec 05 2024
%o A046386 (Python)
%o A046386 from math import isqrt
%o A046386 from sympy import primepi, primerange, integer_nthroot
%o A046386 def A046386(n):
%o A046386     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 A046386     def bisection(f,kmin=0,kmax=1):
%o A046386         while f(kmax) > kmax: kmax <<= 1
%o A046386         while kmax-kmin > 1:
%o A046386             kmid = kmax+kmin>>1
%o A046386             if f(kmid) <= kmid:
%o A046386                 kmax = kmid
%o A046386             else:
%o A046386                 kmin = kmid
%o A046386         return kmax
%o A046386     return bisection(f) # _Chai Wah Wu_, Aug 29 2024
%Y A046386 Products of exactly k distinct primes, for k = 1 to 6: A000040, A006881. A007304, A046386, A046387, A067885.
%Y A046386 Cf. A001221 (omega), A001222 (bigomega), A014613 (bigomega(N) = 4) and A033993 (omega(N) = 4).
%Y A046386 Cf. A046402 (4 palindromic prime factors).
%K A046386 nonn,easy
%O A046386 1,1
%A A046386 _Patrick De Geest_, Jun 15 1998