A069272 11-almost primes (generalization of semiprimes).

2048, 3072, 4608, 5120, 6912, 7168, 7680, 10368, 10752, 11264, 11520, 12800, 13312, 15552, 16128, 16896, 17280, 17408, 17920, 19200, 19456, 19968, 23328, 23552, 24192, 25088, 25344, 25920, 26112, 26880, 28160, 28800, 29184, 29696

Offset: 1

Author

Rick L. Shepherd, Mar 12 2002

Keywords

• nonn

Comments

Product of 11 not necessarily distinct primes.

Divisible by exactly 11 prime powers (not including 1).

Links

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

Crossrefs

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), this sequence (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

Programs

Mathematica

Select[Range[9000], Plus @@ Last /@ FactorInteger[ # ] == 11 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)
Select[Range[30000],PrimeOmega[#]==11&] (* Harvey P. Dale, Jul 13 2013 *)

PARI

k=11; start=2^k; finish=30000; v=[]; for(n=start,finish, if(bigomega(n)==k,v=concat(v,n))); v

PARI

is(n)=bigomega(n)==11 \\ Charles R Greathouse IV, Oct 15 2015

Python

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A069272(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, 11)))
    return bisection(f, n, n) # Chai Wah Wu, Nov 03 2024

Formula

Product p_i^e_i with Sum e_i = 11.

a(n) ~ 3628800n log n / (log log n)^10. - Charles R Greathouse IV, May 06 2013