A285508 Numbers with exactly three prime factors, not all distinct.

8, 12, 18, 20, 27, 28, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 124, 125, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 242, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 343, 356, 363, 369, 387, 388, 404, 412, 423, 425, 428, 436, 452

Offset: 1

Author

Kalle Siukola, Apr 20 2017

Keywords

• easy
• nonn

Comments

Cubes of primes together with products of a prime and the square of a different prime.

Numbers k for which A001222(k) = 3, but A001221(k) < 3. - Antti Karttunen, Apr 20 2017

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Kalle Siukola, Python program

Crossrefs

Cf. A001221, A001222.

Setwise difference of A014612 and A007304.

Union of A030078 and A054753.

Programs

Maple

N:= 1000: # for terms <= N
P:= select(isprime, [2,seq(i,i=3..N/4,2)]): nP:= nops(P):
sort(select(`<=`,[seq(seq(P[i]*P[j]^2,i=1..nP),j=1..nP)],N)); # Robert Israel, Oct 20 2024

Mathematica

Select[Range[452], PrimeOmega[#] == 3 && PrimeNu[#] < 3 &] (* Giovanni Resta, Apr 20 2017 *)

PARI

isA285508(n) = ((omega(n) < 3) && (bigomega(n) == 3));
n=0; k=1; while(k <= 10000, n=n+1; if(isA285508(n),write("b285508.txt", k, " ", n);k=k+1));
\\ Antti Karttunen, Apr 20 2017

Python

from sympy import primefactors, primeomega
def omega(n): return len(primefactors(n))
def bigomega(n): return primeomega(n)
print([n for n in range(1, 501) if omega(n)<3 and bigomega(n) == 3]) # Indranil Ghosh, Apr 20 2017 and Kalle Siukola, Oct 25 2023

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A285508(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**2))-(a<<1)+primepi(isqrt(x//k))-1 for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 20 2024

Scheme

;; With my IntSeq-library.
(define A285508 (MATCHING-POS 1 1 (lambda (n) (and (= 3 (A001222 n)) (< (A001221 n) 3))))) ;; Antti Karttunen, Apr 20 2017