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A069281 20-almost primes (generalization of semiprimes).

%I A069281 #39 Feb 16 2025 08:32:46
%S A069281 1048576,1572864,2359296,2621440,3538944,3670016,3932160,5308416,
%T A069281 5505024,5767168,5898240,6553600,6815744,7962624,8257536,8650752,
%U A069281 8847360,8912896,9175040,9830400,9961472,10223616,11943936,12058624
%N A069281 20-almost primes (generalization of semiprimes).
%C A069281 Product of 20 not necessarily distinct primes.
%C A069281 Divisible by exactly 20 prime powers (not including 1).
%C A069281 Any 20-almost prime can be represented in several ways as a product of two 10-almost primes A046314; in several ways as a product of four 5-almost primes A014614; in several ways as a product of five 4-almost primes A014613; and in several ways as a product of ten semiprimes A001358. - _Jonathan Vos Post_, Dec 12 2004
%H A069281 D. W. Wilson, <a href="/A069281/b069281.txt">Table of n, a(n) for n = 1..10000</a>
%H A069281 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime.</a>
%F A069281 Product p_i^e_i with Sum e_i = 20.
%F A069281 a(n) = A078840(20,n). - _R. J. Mathar_, Jan 30 2019
%t A069281 Select[Range[2*9!,5*10! ],Plus@@Last/@FactorInteger[ # ]==20 &] (* _Vladimir Joseph Stephan Orlovsky_, Feb 26 2009 *)
%o A069281 (PARI) k=20; start=2^k; finish=15000000; v=[]; for(n=start,finish, if(bigomega(n)==k,v=concat(v,n))); v \\ Depending upon the size of k and how many terms are needed, a much more efficient algorithm than the brute-force method above may be desirable. See additional comments in this section of A069280.
%o A069281 (Python)
%o A069281 from math import isqrt, prod
%o A069281 from sympy import primerange, integer_nthroot, primepi
%o A069281 def A069281(n):
%o A069281     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 A069281     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,0,1,1,20)))
%o A069281     kmin, kmax = 1,2
%o A069281     while f(kmax) >= kmax:
%o A069281         kmax <<= 1
%o A069281     while True:
%o A069281         kmid = kmax+kmin>>1
%o A069281         if f(kmid) < kmid:
%o A069281             kmax = kmid
%o A069281         else:
%o A069281             kmin = kmid
%o A069281         if kmax-kmin <= 1:
%o A069281             break
%o A069281     return kmax # _Chai Wah Wu_, Aug 23 2024
%Y A069281 Cf. A101637, A101638, A101605, A101606.
%Y A069281 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), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), this sequence (r = 20). - _Jason Kimberley_, Oct 02 2011
%K A069281 nonn
%O A069281 1,1
%A A069281 _Rick L. Shepherd_, Mar 13 2002