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A162144
Products of cubes of 3 distinct primes.
3
27000, 74088, 287496, 343000, 474552, 1061208, 1157625, 1331000, 1481544, 2197000, 2628072, 3652264, 4492125, 4913000, 5268024, 6028568, 6434856, 6859000, 7414875, 10941048, 12167000, 12326391, 13481272, 14886936, 16581375, 17173512, 18821096
OFFSET
1,1
COMMENTS
Numbers of the form p^3*q^3*r^3 where p, q, r are three distinct primes.
The cubic analog of A085986 (squares of 2 distinct primes).
FORMULA
a(n) = (A007304(n))^3.
A000005(a(n)) = 64.
Sum_{n>=1} 1/a(n) = (P(3)^3 + 2*P(9) - 3*P(3)*P(6))/6 = (A085541^3 + 2*A085969 - 3*A085541*A085966)/6 = 0.0000661486..., where P is the prime zeta function. - Amiram Eldar, Oct 30 2020
EXAMPLE
27000 = 2^3*3^3*5^3. 74088 = 2^3*3^3*7^3. 287496 = 2^3*3^3*11^3.
MATHEMATICA
fQ[n_]:=Last/@FactorInteger[n]=={1, 1, 1}; Select[Range[1000], fQ]^3
PROG
(Python)
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A162144(n):
def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a, k in enumerate(primerange(integer_nthroot(x, 3)[0]+1), 1) for b, m in enumerate(primerange(k+1, isqrt(x//k)+1), a+1)))
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
return bisection(f)**3 # Chai Wah Wu, Aug 30 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by R. J. Mathar, Aug 14 2009
STATUS
approved