A377468 Least perfect-power >= n.

1, 4, 4, 4, 8, 8, 8, 8, 9, 16, 16, 16, 16, 16, 16, 16, 25, 25, 25, 25, 25, 25, 25, 25, 25, 27, 27, 32, 32, 32, 32, 32, 36, 36, 36, 36, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 81, 81, 81

Offset: 1

Author

Gus Wiseman, Nov 05 2024

Keywords

• nonn

Comments

Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.

Crossrefs

The version for prime-powers is A000015.

The union is A001597 (perfect-powers), without powers of two A377702.

Positions of last appearances are also A001597.

The version for primes is A007918 or A151800.

The version for squarefree numbers is A067535.

Run-lengths are A076412.

The opposite version (greatest perfect-power <= n) is A081676.

A000040 lists the primes, differences A001223.

A000961 lists the powers of primes, differences A057820.

A001597 lists the perfect-powers, differences A053289, seconds A376559.

A007916 lists the non-perfect-powers, differences A375706, seconds A376562.

A069623 counts perfect-powers <= n.

A076411 counts perfect-powers < n.

A131605 lists perfect-powers that are not prime-powers.

A377432 counts perfect-powers between primes, zeros A377436.

Cf. A014210, A014234, A023055, A031218, A045542, A052410, A065514, A188951, A216765, A336416, A345531.

Programs

Mathematica

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[NestWhile[#+1&,n,#>1&&!perpowQ[#]&],{n,100}]

Python

from sympy import mobius, integer_nthroot
def A377468(n):
    if n == 1: return 1
    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 = n-f(n-1)
    return bisection(lambda x:f(x)+m,n-1,n) # Chai Wah Wu, Nov 05 2024

Formula

Positions of first appearances for n > 2 are A216765(n-2) = A001597(n-1) + 1.