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A179694
Numbers of the form p^6*q^3 where p and q are distinct primes.
6
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
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
KEYWORD
nonn,changed
AUTHOR
STATUS
approved