A190106 Numbers with prime factorization p^2*q^3*r^3 where p, q, and r are distinct primes.

5400, 9000, 10584, 13500, 24696, 26136, 36504, 37044, 49000, 62424, 68600, 77976, 95832, 114264, 121000, 143748, 158184, 165375, 169000, 171500, 181656, 207576, 231525, 237276, 266200, 289000, 295704, 332024, 353736, 361000, 363096

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, May 04 2011

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Will Nicholes, List of prime signatures, 2010.
• Index to sequences related to prime signature.

Crossrefs

Cf. A179691, A179698, A179746, A189991.

Cf. A085548, A085541, A085965, A085966, A085968.

Programs

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,3,3};Select[Range[500000],f]

PARI

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\4)^(1/6), t1=p^3;forprime(q=p+1, (lim\t1)^(1/3), t2=t1*q^3;forprime(r=2, sqrt(lim\t2), if(p==r||q==r, next);listput(v,t2*r^2)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

Python

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A190106(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((t:=primepi(s:=isqrt(y:=integer_nthroot(x//r**2,3)[0])))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(isqrt(x)+1))+sum(primepi(integer_nthroot(x//p**5,3)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))-primepi(integer_nthroot(x,8)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Formula

Sum_{n>=1} 1/a(n) = P(2)*P(3)^2/2 - P(2)*P(6)/2 - P(3)*P(5) + P(8) = 0.00085907862422456410530..., where P is the prime zeta function. - Amiram Eldar, Mar 07 2024