%I #56 Aug 29 2024 11:12:39
%S 30030,39270,43890,46410,51870,53130,62790,66990,67830,71610,72930,
%T 79170,81510,82110,84630,85470,91770,94710,98670,99330,101010,102102,
%U 103530,106590,108570,110670,111930,114114,115710,117390,122430,123690,124410,125970,128310
%N Products of exactly 6 distinct primes.
%H T. D. Noe, <a href="/A067885/b067885.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F {k: A001221(k) = A001222(k) = 6}. - _R. J. Mathar_, Jul 18 2023
%t Select[Range[125000],PrimeNu[#]==PrimeOmega[#]==6&] (* _Harvey P. Dale_, May 14 2014 *)
%o (PARI) is(n)=factor(n)[,2]==[1,1,1,1,1,1]~ \\ _Charles R Greathouse IV_, Sep 14 2015
%o (PARI) is(n)=omega(n)==6 && bigomega(n)==6 \\ _Hugo Pfoertner_, Dec 18 2018
%o (PARI) list(lim)=lim\=1; my(v=List(), L1,L2,L3,L4,P4,P5); forprime(p=13,lim\2310, L1=lim\p; forprime(q=11,min(L1\210,p-2), L2=L1\q; forprime(r=7, min(L2\30,q-2), L3=L2\r; forprime(s=5,min(L3\6,r-2), L4=L3\s; P4=p*q*r*s; forprime(t=3, min(L4\2,s-2), P5=P4*t; forprime(u=2, min(L4\t,t-1), listput(v,P5*u))))))); Set(v) \\ _Charles R Greathouse IV_, Aug 27 2021
%o (Python)
%o from math import prod, isqrt
%o from sympy import primerange, integer_nthroot, primepi
%o def A067885(n):
%o def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
%o def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,6)))
%o kmin, kmax = 0,1
%o while f(kmax) > kmax:
%o kmax <<= 1
%o while kmax-kmin > 1:
%o kmid = kmax+kmin>>1
%o if f(kmid) <= kmid:
%o kmax = kmid
%o else:
%o kmin = kmid
%o return kmax # _Chai Wah Wu_, Aug 29 2024
%Y Subsequence of A074969. - _R. J. Mathar_, Nov 24 2009
%Y Products of exactly k distinct primes, for k = 1 to 6: A000040, A006881. A007304, A046386, A046387, A067885.
%K nonn
%O 1,1
%A _Benoit Cloitre_, Mar 02 2002