A069273 12-almost primes (generalization of semiprimes).

4096, 6144, 9216, 10240, 13824, 14336, 15360, 20736, 21504, 22528, 23040, 25600, 26624, 31104, 32256, 33792, 34560, 34816, 35840, 38400, 38912, 39936, 46656, 47104, 48384, 50176, 50688, 51840, 52224, 53760, 56320, 57600, 58368, 59392

Offset: 1

Author

Rick L. Shepherd, Mar 13 2002

Keywords

• nonn

Comments

Product of 12 not necessarily distinct primes.

Divisible by exactly 12 prime powers (not including 1).

Any 12-almost prime can be represented in at least one way as a product of two 6-almost primes A046306, three 4-almost primes A014613, four 3-almost primes A014612, or six semiprimes A001358. - Jonathan Vos Post, Dec 11 2004

Links

• D. W. Wilson, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Almost Prime.

Crossrefs

Cf. A101637, A101638, A101605, A101606.

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), this sequence (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

Programs

Mathematica

Select[Range[20000], Plus @@ Last /@ FactorInteger[ # ] == 12 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)
Select[Range[60000],PrimeOmega[#]==12&] (* Harvey P. Dale, May 01 2019 *)

PARI

k=12; start=2^k; finish=70000; v=[]; for(n=start,finish, if(bigomega(n)==k,v=concat(v,n))); v

Python

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A069273(n):
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    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)))
    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, 12)))
    return bisection(f, n, n) # Chai Wah Wu, Nov 03 2024

Formula

Product p_i^e_i with Sum e_i = 12.