A046386 Products of exactly four distinct primes.

210, 330, 390, 462, 510, 546, 570, 690, 714, 770, 798, 858, 870, 910, 930, 966, 1110, 1122, 1155, 1190, 1218, 1230, 1254, 1290, 1302, 1326, 1330, 1365, 1410, 1430, 1482, 1518, 1554, 1590, 1610, 1722, 1770, 1785, 1794, 1806, 1830, 1870, 1914, 1938, 1974

Offset: 1

Author

Patrick De Geest, Jun 15 1998

Keywords

• nonn
• easy

Comments

A squarefree subsequence of A033993. Numbers like 420 = 2^2*3*5*7 with at least one prime exponent greater than 1 in the prime signature are excluded here. - R. J. Mathar, Apr 03 2011

Numbers such that omega(n) = bigomega(n) = 4. - Michel Marcus, Dec 15 2015

Examples

			210 = 2*3*5*7;
330 = 2*3*5*11;
390 = 2*3*5*13;
462 = 2*3*7*11.
		

Links

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

Crossrefs

Products of exactly k distinct primes, for k = 1 to 6: A000040, A006881. A007304, A046386, A046387, A067885.

Cf. A001221 (omega), A001222 (bigomega), A014613 (bigomega(N) = 4) and A033993 (omega(N) = 4).

Cf. A046402 (4 palindromic prime factors).

Programs

Mathematica

fQ[n_] := Last /@ FactorInteger[n] == {1, 1, 1, 1}; Select[ Range[2000], fQ[ # ] &] (* Robert G. Wilson v, Aug 04 2005 *)
Select[Range[2000],PrimeNu[#]==PrimeOmega[#]==4&] (* Harvey P. Dale, Jan 05 2025 *)

PARI

is(n)=factor(n)[,2]==[1,1,1,1]~ \\ Charles R Greathouse IV, Sep 17 2015

PARI

is(n) = omega(n)==4 && bigomega(n)==4 \\ Hugo Pfoertner, Dec 18 2018

PARI

list(lim)=my(v=List()); forprime(p=2,sqrtnint(lim\=1,4), forprime(q=p+1,sqrtnint(lim\p,3), forprime(r=q+2,sqrtint(lim\p\q), my(t=p*q*r); forprime(s=r+2,lim\t, listput(v,t*s))))); Set(v) \\ Charles R Greathouse IV, Dec 05 2024

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A046386(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m*r))-c for a,k in enumerate(primerange(integer_nthroot(x,4)[0]+1),1) for b,m in enumerate(primerange(k+1,integer_nthroot(x//k,3)[0]+1),a+1) for c,r in enumerate(primerange(m+1,isqrt(x//(k*m))+1),b+1)))
    def bisection(f,kmin=0,kmax=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
    return bisection(f) # Chai Wah Wu, Aug 29 2024

Formula

Intersection of A014613 (product of 4 primes) and A033993 (divisible by 4 distinct primes). - M. F. Hasler, Mar 24 2022