A046310 Numbers that are divisible by exactly 8 primes counting multiplicity.

256, 384, 576, 640, 864, 896, 960, 1296, 1344, 1408, 1440, 1600, 1664, 1944, 2016, 2112, 2160, 2176, 2240, 2400, 2432, 2496, 2916, 2944, 3024, 3136, 3168, 3240, 3264, 3360, 3520, 3600, 3648, 3712, 3744, 3968, 4000, 4160, 4374, 4416, 4536, 4704, 4736

Offset: 1

Author

Patrick De Geest, Jun 15 1998

Keywords

• nonn

Comments

Also called 8-almost primes. Products of exactly 8 primes (not necessarily distinct). Any 8-almost prime can be represented in several ways as a product of two 4-almost primes A014613 and in several ways as a product of four semiprimes A001358. - Jonathan Vos Post, Dec 11 2004

Odd terms are in A046321; first odd term is a(64)=6561=3^8. - Zak Seidov, Feb 08 2016

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Reference

Crossrefs

Cf. A046309, A120049 (number of 8-almost primes <= 10^n).

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), this sequence (r = 8), A046312 (r = 9), A046314 (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).

Cf. A046321.

Programs

Maple

A046310 := proc(n)
    option remember;
    if n = 1 then
        2^8 ;
    else
        for a from procname(n-1)+1 do
            if numtheory[bigomega](a) = 8 then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A046310(n),n=1..30) ; # R. J. Mathar, Dec 21 2018

Mathematica

Select[Range[1600], Plus @@ Last /@ FactorInteger[ # ] == 8 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)
Select[Range[5000],PrimeOmega[#]==8&]  (* Harvey P. Dale, Apr 19 2011 *)

PARI

is(n)=bigomega(n)==8 \\ Charles R Greathouse IV, Mar 21 2013

Python

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A046310(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-1+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,8)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

Formula

Product p_i^e_i with Sum e_i = 8.

a(n) ~ 5040n log n / (log log n)^7. - Charles R Greathouse IV, May 06 2013

a(n) = A078840(8,n). - R. J. Mathar, Jan 30 2019