A189988 Numbers with prime factorization p^2*q^4.

144, 324, 400, 784, 1936, 2025, 2500, 2704, 3969, 4624, 5625, 5776, 8464, 9604, 9801, 13456, 13689, 15376, 21609, 21904, 23409, 26896, 29241, 29584, 30625, 35344, 42849, 44944, 55696, 58564, 59536, 60025, 68121, 71824, 75625, 77841

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, May 03 2011

Keywords

• nonn

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
• Index to sequences related to prime signature

Crossrefs

Cf. A178740.

Cf. A085548, A085964, A085966.

Programs

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 *)

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

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