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A069273 12-almost primes (generalization of semiprimes).

%I A069273 #40 Feb 16 2025 08:32:45
%S A069273 4096,6144,9216,10240,13824,14336,15360,20736,21504,22528,23040,25600,
%T A069273 26624,31104,32256,33792,34560,34816,35840,38400,38912,39936,46656,
%U A069273 47104,48384,50176,50688,51840,52224,53760,56320,57600,58368,59392
%N A069273 12-almost primes (generalization of semiprimes).
%C A069273 Product of 12 not necessarily distinct primes.
%C A069273 Divisible by exactly 12 prime powers (not including 1).
%C A069273 Any 12-almost prime can be represented in at least one way as a product of two 6-almost primes A046306, three 4-almost primes A014613, four 3-almost primes A014612, or six semiprimes A001358. - _Jonathan Vos Post_, Dec 11 2004
%H A069273 D. W. Wilson, <a href="/A069273/b069273.txt">Table of n, a(n) for n = 1..10000</a>
%H A069273 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime.</a>
%F A069273 Product p_i^e_i with Sum e_i = 12.
%t A069273 Select[Range[20000], Plus @@ Last /@ FactorInteger[ # ] == 12 &] (* _Vladimir Joseph Stephan Orlovsky_, Apr 23 2008 *)
%t A069273 Select[Range[60000],PrimeOmega[#]==12&] (* _Harvey P. Dale_, May 01 2019 *)
%o A069273 (PARI) k=12; start=2^k; finish=70000; v=[]; for(n=start,finish, if(bigomega(n)==k,v=concat(v,n))); v
%o A069273 (Python)
%o A069273 from math import isqrt, prod
%o A069273 from sympy import primerange, integer_nthroot, primepi
%o A069273 def A069273(n):
%o A069273     def bisection(f, kmin=0, kmax=1):
%o A069273         while f(kmax) > kmax: kmax <<= 1
%o A069273         while kmax-kmin > 1:
%o A069273             kmid = kmax+kmin>>1
%o A069273             if f(kmid) <= kmid:
%o A069273                 kmax = kmid
%o A069273             else:
%o A069273                 kmin = kmid
%o A069273         return kmax
%o A069273     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 A069273     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, 12)))
%o A069273     return bisection(f, n, n) # _Chai Wah Wu_, Nov 03 2024
%Y A069273 Cf. A101637, A101638, A101605, A101606.
%Y A069273 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), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), this sequence (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 A069273 nonn
%O A069273 1,1
%A A069273 _Rick L. Shepherd_, Mar 13 2002