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

1,1

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.

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A179691, A179698, A179746, A189991.

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

Sequence in context: A340109 A364991 A378769 * A252572 A382451 A035902

Adjacent sequences: A190103 A190104 A190105 * A190107 A190108 A190109

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, May 04 2011

Status

approved