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

1,1

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

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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A001221, A001222.

Setwise difference of A014612 and A007304.

Union of A030078 and A054753.

Sequence in context: A378494 A187042 A370650 * A054397 A075818 A090738

Adjacent sequences: A285505 A285506 A285507 * A285509 A285510 A285511

Keyword

easy,nonn

Author

Kalle Siukola, Apr 20 2017

Status

approved