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A069277 16-almost primes (generalization of semiprimes).

%I A069277 #37 Aug 21 2025 18:34:24
%S A069277 65536,98304,147456,163840,221184,229376,245760,331776,344064,360448,
%T A069277 368640,409600,425984,497664,516096,540672,552960,557056,573440,
%U A069277 614400,622592,638976,746496,753664,774144,802816,811008,829440,835584,860160
%N A069277 16-almost primes (generalization of semiprimes).
%C A069277 Product of 16 not necessarily distinct primes.
%C A069277 Divisible by exactly 16 prime powers (not including 1).
%C A069277 Any 16-almost prime can be represented in several ways as a product of two 8-almost primes A046310; in several ways as a product of four 4-almost primes A014613; and in several ways as a product of eight semiprimes A001358. - _Jonathan Vos Post_, Dec 12 2004
%H A069277 D. W. Wilson, <a href="/A069277/b069277.txt">Table of n, a(n) for n = 1..10000</a>
%H A069277 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime.</a>
%F A069277 Product p_i^e_i with Sum e_i = 16.
%t A069277 Select[Range[300000], Plus @@ Last /@ FactorInteger[ # ] == 16 &] (* _Vladimir Joseph Stephan Orlovsky_, Apr 23 2008 *)
%t A069277 Select[Range[10^6],PrimeOmega[#]==16&] (* _Harvey P. Dale_, Jan 30 2015 *)
%o A069277 (PARI) k=16; start=2^k; finish=1000000; v=[]; for(n=start,finish, if(bigomega(n)==k,v=concat(v,n))); v
%o A069277 (Python)
%o A069277 from math import isqrt, prod
%o A069277 from sympy import primerange, integer_nthroot, primepi
%o A069277 def A069277(n):
%o A069277     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 A069277     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,0,1,1,16)))
%o A069277     def bisection(f,kmin=0,kmax=1):
%o A069277         while f(kmax) > kmax: kmax <<= 1
%o A069277         while kmax-kmin > 1:
%o A069277             kmid = kmax+kmin>>1
%o A069277             if f(kmid) <= kmid:
%o A069277                 kmax = kmid
%o A069277             else:
%o A069277                 kmin = kmid
%o A069277         return kmax
%o A069277     return bisection(f) # _Chai Wah Wu_, Aug 31 2024
%Y A069277 Cf. A014610, A101637, A101638, A101605, A101606.
%Y A069277 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), A069275 (r = 14), A069276 (r = 15), this sequence (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - _Jason Kimberley_, Oct 02 2011
%K A069277 nonn,changed
%O A069277 1,1
%A A069277 _Rick L. Shepherd_, Mar 13 2002