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A067885
Products of exactly 6 distinct primes.
32
30030, 39270, 43890, 46410, 51870, 53130, 62790, 66990, 67830, 71610, 72930, 79170, 81510, 82110, 84630, 85470, 91770, 94710, 98670, 99330, 101010, 102102, 103530, 106590, 108570, 110670, 111930, 114114, 115710, 117390, 122430, 123690, 124410, 125970, 128310
OFFSET
1,1
FORMULA
{k: A001221(k) = A001222(k) = 6}. - R. J. Mathar, Jul 18 2023
MATHEMATICA
Select[Range[125000], PrimeNu[#]==PrimeOmega[#]==6&] (* Harvey P. Dale, May 14 2014 *)
PROG
(PARI) is(n)=factor(n)[, 2]==[1, 1, 1, 1, 1, 1]~ \\ Charles R Greathouse IV, Sep 14 2015
(PARI) is(n)=omega(n)==6 && bigomega(n)==6 \\ Hugo Pfoertner, Dec 18 2018
(PARI) list(lim)=lim\=1; my(v=List(), L1, L2, L3, L4, P4, P5); forprime(p=13, lim\2310, L1=lim\p; forprime(q=11, min(L1\210, p-2), L2=L1\q; forprime(r=7, min(L2\30, q-2), L3=L2\r; forprime(s=5, min(L3\6, r-2), L4=L3\s; P4=p*q*r*s; forprime(t=3, min(L4\2, s-2), P5=P4*t; forprime(u=2, min(L4\t, t-1), listput(v, P5*u))))))); Set(v) \\ Charles R Greathouse IV, Aug 27 2021
(Python)
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A067885(n):
def g(x, a, b, c, m): yield from (((d, ) for d in enumerate(primerange(b+1, isqrt(x//c)+1), a+1)) if m==2 else (((a2, b2), )+d for a2, b2 in enumerate(primerange(b+1, integer_nthroot(x//c, m)[0]+1), a+1) 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, 6)))
kmin, kmax = 0, 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 # Chai Wah Wu, Aug 29 2024
CROSSREFS
Subsequence of A074969. - R. J. Mathar, Nov 24 2009
Products of exactly k distinct primes, for k = 1 to 6: A000040, A006881. A007304, A046386, A046387, A067885.
Sequence in context: A285655 A074969 A066765 * A285615 A336671 A258361
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Mar 02 2002
STATUS
approved