A162143 Numbers that are the squares of the product of three distinct primes.

900, 1764, 4356, 4900, 6084, 10404, 11025, 12100, 12996, 16900, 19044, 23716, 27225, 28900, 30276, 33124, 34596, 36100, 38025, 49284, 52900, 53361, 56644, 60516, 65025, 66564, 70756, 74529, 79524, 81225, 81796, 84100, 96100, 101124, 103684, 119025, 125316, 127449

Offset

1,1

Comments

Numbers that are the product of exactly 3 distinct squares of primes (p^2*q^2*r^2).

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Index to sequences related to prime signature

Formula

a(n) = A007304(n)^2.

A050326(a(n)) = 8. - Reinhard Zumkeller, May 03 2013

Sum_{n>=1} 1/a(n) = (P(2)^3 + 2*P(6) - 3*P(2)*P(4))/6 = (A085548^3 + 2*A085966 - 3*A085548*A085964)/6 = 0.0036962441..., where P is the prime zeta function. - Amiram Eldar, Oct 30 2020

Example

900 = 2^2*3^2*5^2, 1764 = 2^2*3^2*7^2, 4356 = 2^2*3^2*11^2, ..

Maple

h := proc(n) local P; P := NumberTheory:-PrimeFactors(n); nops(P) = 3 and n = mul(P) end:
A162143List := upto -> seq(n^2, n=select(h, [seq(1..upto)])):  # Peter Luschny, Apr 14 2025

Mathematica

fQ[n_]:=Last/@FactorInteger[n]=={2, 2, 2}; Select[Range[100000], f]

Prog

(Python)
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A162143(n):
    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), 1) for b, m in enumerate(primerange(k+1, isqrt(x//k)+1), a+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)**2 # Chai Wah Wu, Aug 29 2024
(SageMath)
def is_a(n):
    P = prime_divisors(n)
    return len(P) == 3 and prod(P) == n
print([n*n for n in range(1, 439) if is_a(n)]) # Peter Luschny, Apr 14 2025

Crossrefs

Cf. A007304, A006881, A050326, A162142, A085548, A085964, A085966.

Sequence in context: A074853 A392741 A391320 * A391428 A258888 A287800

Adjacent sequences: A162140 A162141 A162142 * A162144 A162145 A162146

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

Extensions

Edited by N. J. A. Sloane, Jun 27 2009

Status

approved