A143610 Numbers of the form p^2 * q^3, where p,q are distinct primes.

72, 108, 200, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 2312, 2888, 3087, 3267, 4232, 4563, 5324, 6125, 6728, 7688, 7803, 8575, 8788, 9747, 10952, 11979, 13448, 14283, 14792, 15125, 17672, 19652, 19773, 21125, 22472, 22707, 25947, 27436

Offset: 1

Author

M. F. Hasler, Aug 27 2008

Keywords

• easy
• nonn

Comments

Also: numbers with prime signature {3,2}.

This is a subsequence of A114128. [Hasler]

Every a(n) is an Achilles number (A052486). They are minimal, meaning no proper divisor is an Achilles number. - Antonio Roldán, Dec 27 2011

Examples

			The first three terms of this sequence are 3^2 * 2^3 = 72, 2^2 * 3^3 = 108, 5^2 * 2^3 = 200.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Project Euler, Problem 200: Find the 200th prime-proof sqube containing the contiguous sub-string 200.
• Index to sequences related to prime signature

Crossrefs

Cf. A114128.

Cf. A085541, A085548, A085965.

Programs

Mathematica

f[n_] := Sort[Last/@FactorInteger[n]] == {2, 3}; Select[Range[30000], f] (* Vladimir Joseph Stephan Orlovsky, Oct 09 2009 *)

PARI

for(n=1, 10^5, omega(n)==2 || next; vecsort(factor(n)[,2])==[2,3]~ && print1(n","))

PARI

list(lim)=my(v=List(),t);forprime(p=2, (lim\4)^(1/3), t=p^3;forprime(q=2, sqrt(lim\t), if(p==q, next);listput(v,t*q^2))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A143610(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = 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 n+x-sum(primepi(isqrt(x//p**3)) for p in primerange(integer_nthroot(x,3)[0]+1))+primepi(integer_nthroot(x,5)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Formula

Sum_{n>=1} 1/a(n) = P(2)*P(3) - P(5) = A085548 * A085541 - A085965 = 0.043280..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020