OFFSET
1,1
COMMENTS
Numbers k such that tau(k^2)/tau(k) = 3 where tau(n) is the number of divisors of n (A000005). - Michel Marcus, Feb 09 2018
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(2)*P(4) - P(6) = A085548 * A085964 - A085966 = 0.017749..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020
a(n) = A054753(n)^2. - R. J. Mathar, May 05 2023
MATHEMATICA
f[n_]:=Sort[Last/@FactorInteger[n]]=={2, 4}; Select[Range[150000], f]
Module[{upto=80000}, Select[Union[Flatten[{#[[1]]^2 #[[2]]^4, #[[1]]^4 #[[2]]^2}&/@ Subsets[Prime[Range[Sqrt[upto/16]]], {2}]]], #<=upto&]] (* Harvey P. Dale, Dec 15 2017 *)
PROG
(PARI) list(lim)=my(v=List(), t); forprime(p=2, (lim\4)^(1/4), t=p^4; forprime(q=2, sqrt(lim\t), if(p==q, next); listput(v, t*q^2))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011
(Python)
from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A189988(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(isqrt(x//p**4)) for p in primerange(integer_nthroot(x, 4)[0]+1))+primepi(integer_nthroot(x, 6)[0])
return bisection(f, n, n) # Chai Wah Wu, Feb 21 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Joseph Stephan Orlovsky, May 03 2011
STATUS
approved
