A377288 Numbers k such that there are exactly two prime-powers between prime(k)+1 and prime(k+1)-1.

4, 9, 30, 327, 3512

Offset: 1

Author

Gus Wiseman, Oct 25 2024

Keywords

• nonn
• more

Comments

Is this sequence finite? For this conjecture see A053706, A080101, A366833.

Any further terms are > 10^12. - Lucas A. Brown, Nov 08 2024

Examples

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

Links

• Lucas A. Brown, Python program.

Crossrefs

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

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

The corresponding primes are A053706.

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

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

These are the positions of 2 in A080101, or 3 in A366833.

For at least one prime-power we have A377057, primes A053607.

For no prime-powers we have A377286.

For exactly one prime-power we have A377287.

For squarefree instead of prime-power see A377430, A061398, A377431, A068360.

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, A376596, A376597, A377282.

Programs

Mathematica

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==2&]

Formula

prime(a(n)) = A053706(n).