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%I #15 Aug 23 2024 16:31:35
%S 128,192,256,288,320,384,432,448,480,512,576,640,648,672,704,720,768,
%T 800,832,864,896,960,972,1008,1024,1056,1080,1088,1120,1152,1200,1216,
%U 1248,1280,1296,1344,1408,1440,1458,1472,1512,1536,1568,1584,1600,1620
%N Numbers that are divisible by at least 7 primes (counted with multiplicity).
%H Harvey P. Dale, <a href="/A046307/b046307.txt">Table of n, a(n) for n = 1..1000</a>
%F Product p_i^e_i with Sum e_i >= 7.
%t Select[Range[2000],PrimeOmega[#]>6&] (* _Harvey P. Dale_, Nov 16 2012 *)
%o (PARI) is(n)=bigomega(n)>6 \\ _Charles R Greathouse IV_, Sep 17 2015
%o (Python)
%o from math import prod, isqrt
%o from sympy import primerange, integer_nthroot, primepi
%o def A046307(n):
%o def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
%o def f(x): return int(n+primepi(x)+sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(2,7)))
%o kmin, kmax = 1,2
%o while f(kmax) >= kmax:
%o kmax <<= 1
%o while True:
%o kmid = kmax+kmin>>1
%o if f(kmid) < kmid:
%o kmax = kmid
%o else:
%o kmin = kmid
%o if kmax-kmin <= 1:
%o break
%o return kmax # _Chai Wah Wu_, Aug 23 2024
%Y Cf. A046308.
%K nonn
%O 1,1
%A _Patrick De Geest_, Jun 15 1998
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