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A069274 13-almost primes (generalization of semiprimes).

%I A069274 #29 Feb 16 2025 08:32:45
%S A069274 8192,12288,18432,20480,27648,28672,30720,41472,43008,45056,46080,
%T A069274 51200,53248,62208,64512,67584,69120,69632,71680,76800,77824,79872,
%U A069274 93312,94208,96768,100352,101376,103680,104448,107520,112640,115200
%N A069274 13-almost primes (generalization of semiprimes).
%C A069274 Product of 13 not necessarily distinct primes.
%C A069274 Divisible by exactly 13 prime powers (not including 1).
%H A069274 D. W. Wilson, <a href="/A069274/b069274.txt">Table of n, a(n) for n = 1..10000</a>
%H A069274 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime.</a>
%F A069274 Product p_i^e_i with Sum e_i = 13.
%t A069274 Select[Range[30000], Plus @@ Last /@ FactorInteger[ # ] == 13 &] (* _Vladimir Joseph Stephan Orlovsky_, Apr 23 2008 *)
%t A069274 Select[Range[116000],PrimeOmega[#]==13&] (* _Harvey P. Dale_, Mar 11 2019 *)
%o A069274 (PARI) k=13; start=2^k; finish=130000; v=[]; for(n=start,finish, if(bigomega(n)==k,v=concat(v,n))); v
%o A069274 (Python)
%o A069274 from math import isqrt, prod
%o A069274 from sympy import primerange, integer_nthroot, primepi
%o A069274 def A067274(n):
%o A069274     def bisection(f, kmin=0, kmax=1):
%o A069274         while f(kmax) > kmax: kmax <<= 1
%o A069274         while kmax-kmin > 1:
%o A069274             kmid = kmax+kmin>>1
%o A069274             if f(kmid) <= kmid:
%o A069274                 kmax = kmid
%o A069274             else:
%o A069274                 kmin = kmid
%o A069274         return kmax
%o A069274     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 A069274     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, 13)))
%o A069274     return bisection(f, n, n) # _Chai Wah Wu_, Nov 03 2024
%Y A069274 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), this sequence (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 A069274 nonn
%O A069274 1,1
%A A069274 _Rick L. Shepherd_, Mar 13 2002