A377436 Numbers k such that there is no perfect-power x in the range prime(k) < x < prime(k+1).

1, 3, 5, 7, 8, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 26, 27, 28, 29, 32, 33, 35, 36, 37, 38, 40, 41, 42, 43, 45, 46, 49, 50, 51, 52, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 67, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90

Offset: 1

Author

Gus Wiseman, Nov 02 2024

Keywords

• nonn

Comments

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

Examples

			Primes 8 and 9 are 19 and 23, and the interval (20,21,22) contains no prime-powers, so 8 is in the sequence.
		

Crossrefs

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

For squarefree instead of perfect-power we have A068360, see A061398, A377430, A377431.

For just squares (instead of all perfect-powers) we have A221056, primes A224363.

For prime-powers (instead of perfect-powers) we have A377286.

These are the positions of 0 in A377432.

For one instead of none we have A377434, for prime-powers A377287.

For two instead of none we have A377466, for prime-powers A377288, primes A053706.

A000015 gives the least prime-power >= n.

A000040 lists the primes, differences A001223.

A000961 lists the powers of primes, differences A057820.

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

A065514 gives the nearest prime-power before prime(n)-1, difference A377289.

A080101 and A366833 count prime-powers between primes, see A377057, A053607, A304521.

A081676 gives the nearest perfect-power up to n.

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

A377468 gives the nearest perfect-power after n.

Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A276781, A345531, A376597, A377054, A377282.

Programs

Mathematica

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Select[Range[100],Length[Select[Range[Prime[#]+1, Prime[#+1]-1],perpowQ]]==0&]