A378249 Least perfect power > prime(n).

4, 4, 8, 8, 16, 16, 25, 25, 25, 32, 32, 49, 49, 49, 49, 64, 64, 64, 81, 81, 81, 81, 100, 100, 100, 121, 121, 121, 121, 121, 128, 144, 144, 144, 169, 169, 169, 169, 169, 196, 196, 196, 196, 196, 216, 216, 216, 225, 243, 243, 243, 243, 243, 256, 289, 289, 289

Offset

1,1

Comments

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

Which terms appear only once? Just 128, 225, 256, 64009, 1295044?

Links

Michel Marcus, Table of n, a(n) for n = 1..10000

Example

The first number line below shows the perfect powers. The second shows each prime.
-1-----4-------8-9------------16----------------25--27--------32------36------------------------49--
===2=3===5===7======11==13======17==19======23==========29==31==========37======41==43======47======

Mathematica

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[NestWhile[#+1&, Prime[n], radQ[#]&], {n, 100}]

Prog

(PARI) f(p) = p++; while(!ispower(p), p++); p;
lista(nn) = apply(f, primes(nn)); \\ Michel Marcus, Dec 19 2024
(Python)
from sympy import mobius, integer_nthroot, prime
from oeis_sequences.OEISsequences import bisection
def A378249(n):
    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 = (p:=prime(n))-f(p)+1
    return bisection(lambda x:f(x)+m, m, m) # Chai Wah Wu, Dec 20 2025

Crossrefs

A version for prime powers (but starting with prime(k) + 1) is A345531.

Positions of last appearances are A377283, complement A377436.

Restriction of A377468 to the primes, for prime powers A000015.

The opposite is A378035, restriction of A081676.

The union is A378250.

Run lengths are A378251.

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 numbers that are not 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, postpositives A377466.

Cf. A007918, A023055, A031218, A045542, A052410, A065514, A076412, A188951, A216765, A377434.

Sequence in context: A377783 A328527 A389588 * A145447 A204989 A101921

Adjacent sequences: A378246 A378247 A378248 * A378250 A378251 A378252

Keyword

nonn

Author

Gus Wiseman, Nov 21 2024

Status

approved