OFFSET
1,1
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
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
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 *)
PROG
(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
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), 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.
KEYWORD
nonn
AUTHOR
Patrick De Geest, Jun 15 1998
STATUS
approved