OFFSET
1,1
LINKS
Zak Seidov, Table of n, a(n) for n = 1..35737
FORMULA
a(n) ~ A046310(n) ~ 5040n log n / (log log n)^7. - Charles R Greathouse IV, Aug 23 2024
MATHEMATICA
Select[Range[1, 100001, 2], PrimeOmega[#]==8&] (* Harvey P. Dale, Apr 28 2018 *)
PROG
(Python)
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A046321(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, 1, 3, 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
(PARI) list(lim)=my(v=List()); forprime(a=3, lim\2187, my(La=lim\a); forprime(b=3, min(La\729, a), my(Lb=La\b); forprime(c=3, min(Lb\243, b), my(Lc=Lb\c); forprime(d=3, min(Lc\81, c), my(Ld=Lc\d); forprime(e=3, min(Ld\27, d), my(Le=Ld\e, E=a*b*c*d*e); forprime(f=3, min(Le\9, e), my(Lf=Le\f, F=E*f); forprime(g=3, min(Lf\3, f), my(Lg=Lf\g, G=F*g); forprime(h=3, min(Lg, g), listput(v, G*h))))))))); Set(v) \\ Charles R Greathouse IV, Aug 23 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Patrick De Geest, Jun 15 1998
EXTENSIONS
Offset changed 0=>1 by Zak Seidov, Feb 08 2016
STATUS
approved