A123074 -id:A123074 - cp's OEIS frontend

A101605 a(n) = 1 if n is a product of exactly 3 (not necessarily distinct) primes, otherwise 0.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0

Offset: 1

Jonathan Vos Post, Dec 09 2004

			a(28) = 1 because 28 = 2 * 2 * 7 is the product of exactly 3 primes, counted with multiplicity.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n

Cf. A010051, A064911, (char funct. of) A014612, A101637, A123074.

A101605 := proc(n)
    if numtheory[bigomega](n) = 3 then
        1;
    else
        0;
    end if;
end proc: # R. J. Mathar, Mar 13 2015

Table[Boole[PrimeOmega[n] == 3], {n, 100}] (* Jean-François Alcover, Mar 23 2020 *)

is(n)=bigomega(n)==3 \\ Charles R Greathouse IV, Apr 25 2016

a(n) = 1 if n has exactly three prime factors (not necessarily distinct), else a(n) = 0. a(n) = 1 if n is an element of A014612, else a(n) = 0.

a(n) = floor(Omega(n)/3) * floor(3/Omega(n)). - Wesley Ivan Hurt, Jan 10 2013

Description clarified by Antti Karttunen, Jul 23 2017

A123073 Number of ordered triples of primes (p,q,r) such that pqr = n-th 3-almost prime A014612(n).

Original entry on oeis.org

1, 3, 3, 3, 1, 3, 6, 6, 3, 3, 3, 3, 3, 6, 3, 6, 3, 3, 6, 3, 3, 3, 6, 6, 6, 6, 3, 3, 3, 1, 6, 6, 3, 3, 3, 6, 3, 6, 6, 3, 3, 6, 3, 6, 6, 3, 6, 6, 3, 3, 6, 6, 6, 3, 6, 3, 3, 3, 6, 6, 6, 3, 6, 3, 6, 3, 3, 6, 3, 6, 6, 6, 3, 6, 3, 6, 6, 3, 3, 3, 3, 1, 6, 6, 3, 6, 3, 6, 3, 6, 6, 6, 3, 3, 6, 6, 3, 6, 6, 3, 6, 3, 3, 6, 3

Offset: 1

N. J. A. Sloane and T. D. Noe, Sep 29 2006

nonn

The nonzero subsequence of A123074.

Cf. A123074, A014612.

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, primefactors
def A123073(n):
    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
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
    return (1,3,6)[len(primefactors(bisection(f,n,n)))-1] # Chai Wah Wu, Oct 20 2024

More terms from T. D. Noe, Sep 29 2006

cp's OEIS Frontend

A101605 a(n) = 1 if n is a product of exactly 3 (not necessarily distinct) primes, otherwise 0.

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