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A046314 Numbers that are divisible by exactly 10 primes with multiplicity.

%I A046314 #42 Feb 16 2025 08:32:39
%S A046314 1024,1536,2304,2560,3456,3584,3840,5184,5376,5632,5760,6400,6656,
%T A046314 7776,8064,8448,8640,8704,8960,9600,9728,9984,11664,11776,12096,12544,
%U A046314 12672,12960,13056,13440,14080,14400,14592,14848,14976,15872,16000,16640
%N A046314 Numbers that are divisible by exactly 10 primes with multiplicity.
%C A046314 Also called 10-almost primes. Products of exactly 10 primes (not necessarily distinct). Any 10-almost prime can be represented in several ways as a product of two 5-almost primes A014614 and in several ways as a product of five semiprimes A001358. - _Jonathan Vos Post_, Dec 11 2004
%H A046314 T. D. Noe, <a href="/A046314/b046314.txt">Table of n, a(n) for n = 1..10000</a>
%H A046314 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Reference</a>
%F A046314 Product p_i^e_i with Sum e_i = 10.
%F A046314 a(n) ~ 362880n log n / (log log n)^9. - _Charles R Greathouse IV_, May 06 2013
%t A046314 Select[Range[5000], Plus @@ Last /@ FactorInteger[ # ] == 10 &] (* _Vladimir Joseph Stephan Orlovsky_, Apr 23 2008 *)
%t A046314 Select[Range[17000],PrimeOmega[#]==10&] (* _Harvey P. Dale_, Jun 23 2018 *)
%o A046314 (PARI) is(n)=bigomega(n)==10 \\ _Charles R Greathouse IV_, Mar 21 2013
%o A046314 (Python)
%o A046314 from math import isqrt, prod
%o A046314 from sympy import primerange, integer_nthroot, primepi
%o A046314 def A046314(n):
%o A046314     def bisection(f,kmin=0,kmax=1):
%o A046314         while f(kmax) > kmax: kmax <<= 1
%o A046314         while kmax-kmin > 1:
%o A046314             kmid = kmax+kmin>>1
%o A046314             if f(kmid) <= kmid:
%o A046314                 kmax = kmid
%o A046314             else:
%o A046314                 kmin = kmid
%o A046314         return kmax
%o A046314     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 A046314     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,0,1,1,10)))
%o A046314     return bisection(f,n,n) # _Chai Wah Wu_, Nov 03 2024
%Y A046314 Cf. A046313, A120051 (number of 10-almost primes <= 10^n).
%Y A046314 Cf. A101637, A101638, A101605, A101606.
%Y A046314 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), this sequence (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), A069281 (r = 20). - _Jason Kimberley_, Oct 02 2011
%K A046314 nonn
%O A046314 1,1
%A A046314 _Patrick De Geest_, Jun 15 1998