A131605 Perfect powers of nonprimes (m^k where m is a nonprime positive integer and k >= 2).

1, 36, 100, 144, 196, 216, 225, 324, 400, 441, 484, 576, 676, 784, 900, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 1600, 1728, 1764, 1936, 2025, 2116, 2304, 2500, 2601, 2704, 2744, 2916, 3025, 3136, 3249, 3364, 3375, 3600, 3844, 3969, 4225, 4356, 4624

Offset

1,2

Comments

Although 1 is a square, is a cube, and so on..., 1 is sometimes excluded from perfect powers since it is not a well-defined power of 1 (1 = 1^k for any k in [2, 3, 4, 5, ...])

Links

Daniel Forgues, Table of n, a(n) for n=1..8649

Prog

(PARI) isok(n) = if (n == 1, return (1), return (ispower(n, , &np) && (! isprime(np)))); \\ Michel Marcus, Jun 12 2013
(Python)
from sympy import mobius, integer_nthroot, primepi
def A131605(n):
    def f(x): return int(n-2+x+sum(mobius(k)*((a:=integer_nthroot(x, k)[0])-1)+primepi(a) for k in range(2, x.bit_length())))
    kmin, kmax = 1, 2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 14 2024

Crossrefs

Cf. A000961, A001597, A024619, A025475.

Sequence in context: A137945 A250812 A072413 * A296204 A063734 A069057

Adjacent sequences: A131602 A131603 A131604 * A131606 A131607 A131608

Keyword

nonn,changed

Author

Daniel Forgues, May 27 2008

Extensions

Klaus Brockhaus previously provided a table of n, a(n) for n=1..1323, May 28 2008

Status

approved