A377286 Numbers k such that there are no prime-powers between prime(k)+1 and prime(k+1)-1.

1, 3, 5, 7, 8, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82

Offset: 1

Author

Gus Wiseman, Oct 25 2024

Keywords

• nonn

Examples

			Primes 18 and 19 are 61 and 67, and the interval (62, 63, 64, 65, 66) contains the prime-power 64, so 18 is not in the sequence.
		

Crossrefs

The interval from A008864(n) to A006093(n+1) has A046933(n) elements.

For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.

The nearest prime-power before prime(n)-1 is A065514, difference A377289.

These are the positions of 0 in A080101, or 1 in A366833.

The nearest prime-power after prime(n)+1 is A345531, difference A377281.

For at least one prime-power we have A377057.

For one instead of no prime-powers we have A377287.

For two instead of no prime-powers we have A377288.

A000015 gives the least prime-power >= n.

A000040 lists the primes, differences A001223.

A000961 lists the powers of primes, differences A057820.

A031218 gives the greatest prime-power <= n.

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

Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A095195, A276781, A376597, A377051, A377282.

Programs

Mathematica

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==0&]

Python

from itertools import count, islice
from sympy import factorint, nextprime
def A377286_gen(): # generator of terms
    p, q, k = 2, 3, 1
    for k in count(1):
        if all(len(factorint(i))>1 for i in range(p+1,q)):
            yield k
        p, q = q, nextprime(q)
A377286_list = list(islice(A377286_gen(),66)) # Chai Wah Wu, Oct 27 2024