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A046308 Numbers that are divisible by exactly 7 primes counting multiplicity.

%I A046308 #41 Feb 16 2025 08:32:39
%S A046308 128,192,288,320,432,448,480,648,672,704,720,800,832,972,1008,1056,
%T A046308 1080,1088,1120,1200,1216,1248,1458,1472,1512,1568,1584,1620,1632,
%U A046308 1680,1760,1800,1824,1856,1872,1984,2000,2080,2187,2208,2268,2352,2368,2376
%N A046308 Numbers that are divisible by exactly 7 primes counting multiplicity.
%C A046308 Also called 7-almost primes. Products of exactly 7 primes (not necessarily distinct). - _Jonathan Vos Post_, Dec 11 2004
%C A046308 Also, positions of 7 in A001222. - _Zak Seidov_, Oct 14 2012
%H A046308 T. D. Noe, <a href="/A046308/b046308.txt">Table of n, a(n) for n = 1..10000</a>
%H A046308 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Reference</a>
%F A046308 Product p_i^e_i with sum e_i = 7.
%F A046308 a(n) ~ 720n log n / (log log n)^6. - _Charles R Greathouse IV_, May 06 2013
%F A046308 a(n) = A078840(7,n). - _R. J. Mathar_, Jan 30 2019
%t A046308 Select[Range[900], Plus @@ Last /@ FactorInteger[ # ] == 7 &] (* _Vladimir Joseph Stephan Orlovsky_, Apr 23 2008 *)
%o A046308 (PARI) is(n)=bigomega(n)==7 \\ _Charles R Greathouse IV_, Mar 21 2013
%o A046308 (Python)
%o A046308 from math import prod, isqrt
%o A046308 from sympy import primerange, integer_nthroot, primepi
%o A046308 def A046308(n):
%o A046308     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 A046308     def f(x): return int(n-1+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,7)))
%o A046308     kmin, kmax = 1,2
%o A046308     while f(kmax) >= kmax:
%o A046308         kmax <<= 1
%o A046308     while True:
%o A046308         kmid = kmax+kmin>>1
%o A046308         if f(kmid) < kmid:
%o A046308             kmax = kmid
%o A046308         else:
%o A046308             kmin = kmid
%o A046308         if kmax-kmin <= 1:
%o A046308             break
%o A046308     return kmax # _Chai Wah Wu_, Aug 23 2024
%Y A046308 Cf. A120048 (number of 7-almost primes <= 10^n).
%Y A046308 Cf. A101605, A101606, A001222.
%Y A046308 Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), this sequence (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - _Jason Kimberley_, Oct 02 2011
%K A046308 nonn
%O A046308 1,1
%A A046308 _Patrick De Geest_, Jun 15 1998