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A069275 14-almost primes (generalization of semiprimes).

%I A069275 #32 Feb 16 2025 08:32:45
%S A069275 16384,24576,36864,40960,55296,57344,61440,82944,86016,90112,92160,
%T A069275 102400,106496,124416,129024,135168,138240,139264,143360,153600,
%U A069275 155648,159744,186624,188416,193536,200704,202752,207360,208896,215040,225280
%N A069275 14-almost primes (generalization of semiprimes).
%C A069275 Product of 14 not necessarily distinct primes.
%C A069275 Divisible by exactly 14 prime powers (not including 1).
%C A069275 Any 14-almost prime can be represented in several ways as a product of two 7-almost primes A046308; and in several ways as a product of seven semiprimes A001358. - _Jonathan Vos Post_, Dec 11 2004
%H A069275 D. W. Wilson, <a href="/A069275/b069275.txt">Table of n, a(n) for n = 1..10000</a>
%H A069275 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime.</a>
%F A069275 Product p_i^e_i with Sum e_i = 14.
%t A069275 Select[Range[50000], Plus @@ Last /@ FactorInteger[ # ] == 14 &] (* _Vladimir Joseph Stephan Orlovsky_, Apr 23 2008 *)
%o A069275 (PARI) k=14; start=2^k; finish=240000; v=[]; for(n=start,finish, if(bigomega(n)==k,v=concat(v,n))); v
%o A069275 (Python)
%o A069275 from math import isqrt, prod
%o A069275 from sympy import primerange, integer_nthroot, primepi
%o A069275 def A069275(n):
%o A069275     def bisection(f,kmin=0,kmax=1):
%o A069275         while f(kmax) > kmax: kmax <<= 1
%o A069275         while kmax-kmin > 1:
%o A069275             kmid = kmax+kmin>>1
%o A069275             if f(kmid) <= kmid:
%o A069275                 kmax = kmid
%o A069275             else:
%o A069275                 kmin = kmid
%o A069275         return kmax
%o A069275     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 A069275     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,0,1,1,14)))
%o A069275     return bisection(f,n,n) # _Chai Wah Wu_, Nov 03 2024
%Y A069275 Cf. A101637, A101638, A101605, A101606.
%Y A069275 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), A069273 (r = 12), A069274 (r = 13), this sequence(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 A069275 nonn
%O A069275 1,1
%A A069275 _Rick L. Shepherd_, Mar 13 2002