A377434 Numbers k such that there is a unique perfect-power x in the range prime(k) < x < prime(k+1).

2, 6, 15, 18, 22, 25, 31, 34, 39, 44, 47, 48, 53, 54, 61, 66, 68, 72, 78, 85, 92, 97, 99, 105, 114, 122, 129, 137, 146, 154, 162, 168, 172, 181, 191, 200, 210, 217, 219, 228, 240, 251, 263, 269, 274, 283, 295, 306, 309, 319, 329, 342, 357, 367, 378, 393, 400

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 4 and 5 are 7 and 11, and the interval (8,9,10) contains two perfect-powers (8,9), so 4 is not in the sequence.
Primes 5 and 6 are 11 and 13, and the interval (12) contains no perfect-powers, so 5 is not in the sequence.
Primes 6 and 7 are 13 and 17, and the interval (14,15,16) contains just one perfect-power (16), so 6 is in the sequence.
		

Crossrefs

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

For prime-powers we have A377287.

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

These are the positions of 1 in A377432.

For no perfect-powers we have A377436.

For more than one perfect-power we have A377466.

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.

A031218 gives the greatest prime-power <= n.

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

A065514 gives the greatest prime-power < prime(n), difference A377289.

A081676 gives the greatest perfect-power <= n.

A131605 lists perfect-powers that are not prime-powers.

A345531 gives the least prime-power > prime(n), difference A377281.

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

A377468 gives the least perfect-power > n.

Cf. A006549, A023055, A045542, A052410, A069623, A080101, A216765, A224363, A246655, A336416, A375740, A376560, A376561, A377057.

Programs

Mathematica

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