OFFSET
1,1
COMMENTS
Also called 9-almost primes. Products of exactly 9 primes (not necessarily distinct). - 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, Reference
FORMULA
Product p_i^e_i with Sum e_i = 9.
a(n) ~ 40320n log n / (log log n)^8. - Charles R Greathouse IV, May 06 2013
MATHEMATICA
Select[Range[2200], Plus @@ Last /@ FactorInteger[ # ] == 9 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)
Select[Range[10000], PrimeOmega[#]==9&] (* Harvey P. Dale, Oct 24 2020 *)
PROG
(PARI) is(n)=bigomega(n)==9 \\ Charles R Greathouse IV, Mar 21 2013
(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A046312(n):
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
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, 9)))
return bisection(f, n, n) # Chai Wah Wu, Nov 03 2024
CROSSREFS
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), this sequence (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
KEYWORD
nonn
AUTHOR
Patrick De Geest, Jun 15 1998
STATUS
approved