A377466 Numbers k such that there is more than one perfect power x in the range prime(k) < x < prime(k+1).

4, 9, 11, 30, 327, 445, 3512, 7789, 9361, 26519413

Offset: 1

Author

Gus Wiseman, Nov 02 2024

Keywords

• nonn
• more

Comments

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

Is this sequence finite?

The Redmond-Sun conjecture (see A308658) implies that this sequence is finite. - Pontus von Brömssen, Nov 05 2024

Examples

			Primes 9 and 10 are 23 and 29, and the interval (24,25,26,27,28) contains two perfect powers (25,27), so 9 is in the sequence.
		

Crossrefs

For powers of 2 see A013597, A014210, A014234, A188951, A244508, A377467.

For no prime-powers we have A377286, ones in A080101.

For a unique prime-power we have A377287.

For squarefree numbers see A377430, A061398, A377431, A068360, A224363.

These are the positions of terms > 1 in A377432.

For a unique perfect power we have A377434.

For no perfect powers we have A377436.

A000015 gives the least prime power >= n.

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.

A046933 counts the interval from A008864(n) to A006093(n+1).

A081676 gives the greatest perfect power <= n.

A131605 lists perfect powers that are not prime-powers.

A246655 lists the prime-powers not including 1, complement A361102.

A366833 counts prime-powers between primes, see A053607, A304521.

A377468 gives the least perfect power > n.

Cf. A000720, A023055, A031218, A045542, A052410, A053706, A069623, A116086, A116455, A216765, A308658, A336416, A345531, A375740, A376560, A376561, A377057.

Programs

Mathematica

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Select[Range[100],Count[Range[Prime[#]+1, Prime[#+1]-1],_?perpowQ]>1&]

Python

from itertools import islice
from sympy import prime
from gmpy2 import is_power, next_prime
def A377466_gen(startvalue=1): # generator of terms >= startvalue
    k = max(startvalue,1)
    p = prime(k)
    while (q:=next_prime(p)):
        c = 0
        for i in range(p+1,q):
            if is_power(i):
                c += 1
                if c>1:
                    yield k
                    break
        k += 1
        p = q
A377466_list = list(islice(A377466_gen(),9)) # Chai Wah Wu, Nov 04 2024

Formula

a(n) = A000720(A116086(n)) = A000720(A116455(n)) for n <= 10. This would hold for all n if there do not exist more than two perfect powers between any two consecutive primes, which is implied by the Redmond-Sun conjecture. - Pontus von Brömssen, Nov 05 2024

Extensions

a(10) from Pontus von Brömssen, Nov 04 2024