A046308 Numbers that are divisible by exactly 7 primes counting multiplicity.

128, 192, 288, 320, 432, 448, 480, 648, 672, 704, 720, 800, 832, 972, 1008, 1056, 1080, 1088, 1120, 1200, 1216, 1248, 1458, 1472, 1512, 1568, 1584, 1620, 1632, 1680, 1760, 1800, 1824, 1856, 1872, 1984, 2000, 2080, 2187, 2208, 2268, 2352, 2368, 2376

Offset

1,1

Comments

Also called 7-almost primes. Products of exactly 7 primes (not necessarily distinct). - Jonathan Vos Post, Dec 11 2004

Also, positions of 7 in A001222. - Zak Seidov, Oct 14 2012

Links

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

Eric Weisstein's World of Mathematics, Reference

Formula

Product p_i^e_i with sum e_i = 7.

a(n) ~ 720n log n / (log log n)^6. - Charles R Greathouse IV, May 06 2013

a(n) = A078840(7,n). - R. J. Mathar, Jan 30 2019

Mathematica

Select[Range[900], Plus @@ Last /@ FactorInteger[ # ] == 7 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)

Prog

(PARI) is(n)=bigomega(n)==7 \\ Charles R Greathouse IV, Mar 21 2013
(Python)
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A046308(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, 7)))
    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. A120048 (number of 7-almost primes <= 10^n).

Cf. A101605, A101606, A001222.

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), this sequence (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: A114418 A046307 A036331 * A110290 A045028 A378374

Adjacent sequences: A046305 A046306 A046307 * A046309 A046310 A046311

Keyword

nonn,changed

Author

Patrick De Geest, Jun 15 1998

Status

approved