A376120 Refactorable numbers that are perfect powers.

1, 8, 9, 36, 128, 225, 441, 625, 1089, 1521, 2025, 2601, 3249, 3600, 4761, 5625, 6561, 7569, 7776, 8100, 8649, 10000, 12321, 15129, 16641, 19881, 21952, 22500, 25281, 26244, 28224, 31329, 32400, 32768, 33489, 35721, 40401, 45369, 47961, 50625, 56169, 62001, 64000, 71289, 84681, 90000

Offset

1,2

Comments

Intersection of A001597 and A033950.

Links

Table of n, a(n) for n=1..46.

Example

8 is a perfect power, as 8=2^3, and it is also a refactorable numbers, being divisible by its number of divisors (4).

Mathematica

Join[{1}, Select[Range[10^5], Divisible[#, DivisorSigma[0, #]]&&GCD@@FactorInteger[#][[All, 2]]>1&]]

Prog

(PARI) ok(n) = n==1 || (n%numdiv(n)==0&&ispower(n))
(Python)
from itertools import count, islice
from math import prod
from sympy import mobius, integer_nthroot, factorint
def A376120_gen(): # generator of terms
    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
    def f(x): return int(x-1+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))
    m = 1
    for n in count(1):
        m = bisection(lambda x:f(x)+n, m, m)
        if not m%prod(e+1 for e in factorint(m).values()): yield m
A376120_list = list(islice(A376120_gen(), 40)) # Chai Wah Wu, Oct 04 2024

Crossrefs

Cf. A001597 (perfect powers), A033950 (refactorable numbers).

Sequence in context: A322797 A075079 A317379 * A298665 A284822 A222354

Adjacent sequences: A376117 A376118 A376119 * A376121 A376122 A376123

Keyword

nonn

Author

Waldemar Puszkarz, Sep 11 2024

Status

approved