A123074 -id:A123074

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A101605

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

+10
27

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

FORMULA

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

EXAMPLE

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

MAPLE

A101605 := proc(n)

if numtheory[bigomega](n) = 3 then

else

end if;

end proc: # R. J. Mathar, Mar 13 2015

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Dec 09 2004

EXTENSIONS

Description clarified by Antti Karttunen, Jul 23 2017

STATUS

approved

A123073

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The nonzero subsequence of A123074.

LINKS

Table of n, a(n) for n=1..105.

PROG

(Python)

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

CROSSREFS

Cf. A123074, A014612.

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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