A067885 Products of exactly 6 distinct primes.

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

Author

Benoit Cloitre, Mar 02 2002

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Index to sequences related to prime signature

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.

Programs

Mathematica

Select[Range[125000],PrimeNu[#]==PrimeOmega[#]==6&] (* Harvey P. Dale, May 14 2014 *)

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

Formula

{k: A001221(k) = A001222(k) = 6}. - R. J. Mathar, Jul 18 2023