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

1,1

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Index to sequences related to prime signature

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

Adjacent sequences: A067882 A067883 A067884 * A067886 A067887 A067888

Keyword

nonn

Author

Benoit Cloitre, Mar 02 2002

Status

approved