A189990 Numbers with prime factorization p^2*q^6.

576, 1600, 2916, 3136, 7744, 10816, 18225, 18496, 23104, 33856, 35721, 53824, 61504, 62500, 87616, 88209, 107584, 118336, 123201, 140625, 141376, 179776, 210681, 222784, 238144, 263169, 287296, 322624, 341056, 385641, 399424, 440896, 470596

Offset

1,1

Links

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

Will Nicholes, Prime Signatures

Index to sequences related to prime signature

Formula

Sum_{n>=1} 1/a(n) = P(2)*P(6) - P(8) = A085548 * A085966 - A085968 = 0.003658..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={2, 6}; Select[Range[800000], f]

Prog

(PARI) list(lim)=my(v=List(), t); forprime(p=2, (lim\4)^(1/6), t=p^6; forprime(q=2, sqrt(lim\t), if(p==q, next); listput(v, t*q^2))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011
(Python)
from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A189990(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**6)) for p in primerange(integer_nthroot(x, 6)[0]+1))+primepi(integer_nthroot(x, 8)[0])
    return bisection(f, n, n) # Chai Wah Wu, Feb 21 2025

Crossrefs

Subsequence of A137484.

Cf. A179645, A179664.

Cf. A085548, A085966, A085968.

Sequence in context: A268773 A250789 A137484 * A064254 A281242 A067225

Adjacent sequences: A189987 A189988 A189989 * A189991 A189992 A189993

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, May 03 2011

Status

approved