A069277 16-almost primes (generalization of semiprimes).

65536, 98304, 147456, 163840, 221184, 229376, 245760, 331776, 344064, 360448, 368640, 409600, 425984, 497664, 516096, 540672, 552960, 557056, 573440, 614400, 622592, 638976, 746496, 753664, 774144, 802816, 811008, 829440, 835584, 860160

Offset: 1

Author

Rick L. Shepherd, Mar 13 2002

Keywords

• nonn
• changed

Comments

Product of 16 not necessarily distinct primes.

Divisible by exactly 16 prime powers (not including 1).

Any 16-almost prime can be represented in several ways as a product of two 8-almost primes A046310; in several ways as a product of four 4-almost primes A014613; and in several ways as a product of eight semiprimes A001358. - Jonathan Vos Post, Dec 12 2004

Links

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

Crossrefs

Cf. A014610, 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), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), this sequence (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011

Programs

Mathematica

Select[Range[300000], Plus @@ Last /@ FactorInteger[ # ] == 16 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)
Select[Range[10^6],PrimeOmega[#]==16&] (* Harvey P. Dale, Jan 30 2015 *)

PARI

k=16; start=2^k; finish=1000000; 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 A069277(n):
    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,0,1,1,16)))
    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
    return bisection(f) # Chai Wah Wu, Aug 31 2024

Formula

Product p_i^e_i with Sum e_i = 16.