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A046321 Odd numbers divisible by exactly 8 primes (counted with multiplicity).

%I A046321 #18 Aug 23 2024 19:58:34
%S A046321 6561,10935,15309,18225,24057,25515,28431,30375,35721,37179,40095,
%T A046321 41553,42525,47385,50301,50625,56133,59535,61965,63423,66339,66825,
%U A046321 67797,69255,70875,78975,80919,83349,83835,84375,86751,88209,89667
%N A046321 Odd numbers divisible by exactly 8 primes (counted with multiplicity).
%H A046321 Zak Seidov, <a href="/A046321/b046321.txt">Table of n, a(n) for n = 1..35737</a>
%F A046321 a(n) ~ A046310(n) ~ 5040n log n / (log log n)^7. - _Charles R Greathouse IV_, Aug 23 2024
%t A046321 Select[Range[1,100001,2],PrimeOmega[#]==8&] (* _Harvey P. Dale_, Apr 28 2018 *)
%o A046321 (Python)
%o A046321 from math import prod, isqrt
%o A046321 from sympy import primerange, integer_nthroot, primepi
%o A046321 def A046321(n):
%o A046321     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)))
%o A046321     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)))
%o A046321     kmin, kmax = 1,2
%o A046321     while f(kmax) >= kmax:
%o A046321         kmax <<= 1
%o A046321     while True:
%o A046321         kmid = kmax+kmin>>1
%o A046321         if f(kmid) < kmid:
%o A046321             kmax = kmid
%o A046321         else:
%o A046321             kmin = kmid
%o A046321         if kmax-kmin <= 1:
%o A046321             break
%o A046321     return kmax # _Chai Wah Wu_, Aug 23 2024
%o A046321 (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
%Y A046321 Cf. A046310.
%K A046321 nonn
%O A046321 1,1
%A A046321 _Patrick De Geest_, Jun 15 1998
%E A046321 Offset changed 0=>1 by _Zak Seidov_, Feb 08 2016