A179694 Numbers of the form p^6*q^3 where p and q are distinct primes.

1728, 5832, 8000, 21952, 85184, 91125, 125000, 140608, 250047, 314432, 421875, 438976, 778688, 941192, 970299, 1560896, 1601613, 1906624, 3176523, 3241792, 3581577, 4410944, 5000211, 5088448, 5359375, 6644672

Offset

1,1

Links

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

Will Nicholes, Prime Signatures

Formula

Sum_{n>=1} 1/a(n) = P(3)*P(6) - P(9) = A085541 * A085966 - A085969 = 0.000978..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

a(n) = A054753(n)^3. - R. J. Mathar, May 05 2023

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={3, 6}; Select[Range[10^6], f]

Prog

(PARI) list(lim)=my(v=List(), t); forprime(p=2, (lim\8)^(1/6), t=p^6; forprime(q=2, (lim\t)^(1/3), if(p==q, next); listput(v, t*q^3))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011
(Python)
from sympy import primepi, integer_nthroot, primerange
def A179694(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(integer_nthroot(x//p**6, 3)[0]) for p in primerange(integer_nthroot(x, 6)[0]+1))+primepi(integer_nthroot(x, 9)[0])
    return bisection(f, n, n) # Chai Wah Wu, Feb 21 2025

Crossrefs

Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688, A179689, A179690, A179691, A179692, A179693.

Cf. A085541, A085966, A085969.

Sequence in context: A032354 A305475 A202458 * A202200 A251188 A323560

Adjacent sequences: A179691 A179692 A179693 * A179695 A179696 A179697

Keyword

nonn,changed

Author

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

Status

approved