A046306 Numbers that are divisible by exactly 6 primes with multiplicity.

64, 96, 144, 160, 216, 224, 240, 324, 336, 352, 360, 400, 416, 486, 504, 528, 540, 544, 560, 600, 608, 624, 729, 736, 756, 784, 792, 810, 816, 840, 880, 900, 912, 928, 936, 992, 1000, 1040, 1104, 1134, 1176, 1184, 1188, 1215, 1224, 1232, 1260, 1312, 1320

Offset

1,1

Comments

Also called 6-almost primes. Products of exactly 6 primes (not necessarily distinct). Any 6-almost prime can be represented in several ways as a product of two 3-almost primes A014612 and in several ways as a product of three semiprimes A001358. - Jonathan Vos Post, Dec 11 2004

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Almost Prime

Formula

Product p_i^e_i with Sum e_i = 6.

a(n) ~ 120n log n / (log log n)^5. - Charles R Greathouse IV, May 06 2013

a(n) = A078840(6,n). - R. J. Mathar, Jan 30 2019

Mathematica

Select[Range[500], Plus @@ Last /@ FactorInteger[ # ] == 6 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)
Select[Range[1400], PrimeOmega[#]==6&] (* Harvey P. Dale, May 21 2012 *)

Prog

(PARI) is(n)=bigomega(n)==6 \\ Charles R Greathouse IV, Mar 21 2013
(Python)
from math import isqrt, prod
from sympy import primepi, primerange, integer_nthroot
def A046306(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, 6)))
    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

Crossrefs

Cf. A046305, A120047 (number of 6-almost primes <= 10^n).

Cf. 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), this sequence (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), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011

Sequence in context: A046305 A114828 A036330 * A223086 A175163 A111730

Adjacent sequences: A046303 A046304 A046305 * A046307 A046308 A046309

Keyword

nonn

Author

Patrick De Geest, Jun 15 1998

Status

approved