A046309 Numbers that are divisible by at least 8 primes (counted with multiplicity).

256, 384, 512, 576, 640, 768, 864, 896, 960, 1024, 1152, 1280, 1296, 1344, 1408, 1440, 1536, 1600, 1664, 1728, 1792, 1920, 1944, 2016, 2048, 2112, 2160, 2176, 2240, 2304, 2400, 2432, 2496, 2560, 2592, 2688, 2816, 2880, 2916, 2944, 3024, 3072, 3136

Offset

1,1

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Formula

Product p_i^e_i with Sum e_i >= 8.

Mathematica

Select[Range[3200], PrimeOmega[#]>7&] (* Harvey P. Dale, May 29 2013 *)

Prog

(PARI) is(n)=bigomega(n)>7 \\ Charles R Greathouse IV, Sep 17 2015
(Python)
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A046309(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+primepi(x)+sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x, 0, 1, 1, i)) for i in range(2, 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

Crossrefs

Cf. A046310.

Sequence in context: A345534 A345786 A186473 * A036332 A114987 A046310

Adjacent sequences: A046306 A046307 A046308 * A046310 A046311 A046312

Keyword

nonn

Author

Patrick De Geest, Jun 15 1998

Status

approved