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A069276 15-almost primes (generalization of semiprimes).

%I A069276 #34 Feb 16 2025 08:32:45
%S A069276 32768,49152,73728,81920,110592,114688,122880,165888,172032,180224,
%T A069276 184320,204800,212992,248832,258048,270336,276480,278528,286720,
%U A069276 307200,311296,319488,373248,376832,387072,401408,405504,414720,417792,430080
%N A069276 15-almost primes (generalization of semiprimes).
%C A069276 Product of 15 not necessarily distinct primes.
%C A069276 Divisible by exactly 15 prime powers (not including 1).
%C A069276 Any 15-almost prime can be represented in several ways as a product of three 5-almost primes A014614, and in several ways as a product of five 3-almost primes A014612. - _Jonathan Vos Post_, Dec 11 2004
%H A069276 D. W. Wilson, <a href="/A069276/b069276.txt">Table of n, a(n) for n = 1..10000</a>
%H A069276 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime.</a>
%F A069276 Product p_i^e_i with Sum e_i = 15.
%t A069276 Select[Range[90000], Plus @@ Last /@ FactorInteger[ # ] == 15 &] (* _Vladimir Joseph Stephan Orlovsky_, Apr 23 2008 *)
%t A069276 Select[Range[450000],PrimeOmega[#]==15&] (* _Harvey P. Dale_, Aug 14 2019 *)
%o A069276 (PARI) k=15; start=2^k; finish=500000; v=[]; for(n=start,finish, if(bigomega(n)==k,v=concat(v,n))); v
%o A069276 (Python)
%o A069276 from math import isqrt, prod
%o A069276 from sympy import primerange, integer_nthroot, primepi
%o A069276 def A069276(n):
%o A069276     def bisection(f,kmin=0,kmax=1):
%o A069276         while f(kmax) > kmax: kmax <<= 1
%o A069276         while kmax-kmin > 1:
%o A069276             kmid = kmax+kmin>>1
%o A069276             if f(kmid) <= kmid:
%o A069276                 kmax = kmid
%o A069276             else:
%o A069276                 kmin = kmid
%o A069276         return kmax
%o A069276     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 A069276     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,0,1,1,15)))
%o A069276     return bisection(f,n,n) # _Chai Wah Wu_, Nov 03 2024
%Y A069276 Cf. A101637, A101638, A101605, A101606.
%Y A069276 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), this sequence (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - _Jason Kimberley_, Oct 02 2011
%K A069276 nonn
%O A069276 1,1
%A A069276 _Rick L. Shepherd_, Mar 13 2002