A377057 Numbers k such that there is at least one prime-power between prime(k)+1 and prime(k+1)-1.

2, 4, 6, 9, 11, 15, 18, 22, 30, 31, 39, 53, 54, 61, 68, 72, 97, 99, 114, 129, 146, 162, 172, 217, 219, 263, 283, 309, 327, 329, 357, 409, 445, 487, 519, 564, 609, 656, 675, 705, 811, 847, 882, 886, 1000, 1028, 1163, 1252, 1294, 1381, 1423, 1457

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 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 corresponding primes are A053607.

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

These are the positions of positive terms in A080101, or terms >1 in A366833.

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

For no prime-powers we have A377286.

For exactly one prime-power we have A377287.

For exactly two prime-powers we have A377288, primes A053706.

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, A224363, A276781, A376597, A377051, A377282.

Programs

Mathematica

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

Python

from itertools import count, islice
from sympy import factorint, nextprime
def A377057_gen(): # generator of terms
    p, q, k = 2, 3, 1
    for k in count(1):
        if any(len(factorint(i))<=1 for i in range(p+1,q)):
            yield k
        p, q = q, nextprime(q)
A377057_list = list(islice(A377057_gen(),52)) # Chai Wah Wu, Oct 27 2024

Formula

prime(a(n)) = A053607(n).