A162143 a(n) = A007304(n)^2.

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

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

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, ..

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

Crossrefs

Cf. A006881, A050326, A162142.

Cf. A085548, A085964, A085966.

Sequence in context: A321206 A336680 A074853 * A258888 A287800 A246887

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